This paper investigates generating symmetric trajectories for an underactuated biped during the stance phase of running. We use a point mass biped (PMB) model for gait analysis that consists of a prismatic force actuator on a massless leg. The significance of this model is its ability to generate more general and versatile running gaits than the spring-loaded inverted pendulum (SLIP) model, making it more suitable as a template for real robots. The algorithm plans the necessary leg actuator force to cause the robot center of mass to undergo arbitrary trajectories in stance with any arbitrary attack angle and velocity angle. The necessary actuator forces follow from the inverse kinematics and dynamics. Then these calculated forces become the control input to the dynamic model. We compare various center-of-mass trajectories, including a circular arc and polynomials of the degrees 2, 4 and 6. The cost of transport and maximum leg force are calculated for various attack angles and velocity angles. The results show that choosing the velocity angle as small as possible is beneficial, but the angle of attack has an optimum value. We also find a new result: there exist biped running gaits with double-hump ground reaction force profiles which result in less maximum leg force than single-hump profiles.
Citation: Dadashzadeh B, Esmaeili M, Macnab C (2017) Arbitrary Symmetric Running Gait Generation for an Underactuated Biped Model. PLoS ONE 12(1): e0170122. https://doi.org/10.1371/journal.pone.0170122
Editor: Johnny Padulo, University e-Campus, ITALY
Received: July 12, 2016; Accepted: December 29, 2016; Published: January 24, 2017
Copyright: © 2017 Dadashzadeh et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: All relevant data are within the paper.
Funding: The authors received no specific funding for this work.
Competing interests: The authors have declared that no competing interests exist.
The field of legged locomotion has proposed some fundamental models, including the Spring Loaded Inverted Pendulum (SLIP), the active SLIP, and the Point Mass Biped (PMB). These models allow one to investigate both walking and running gaits. The SLIP model remains a popular tool to investigate bipedal walking and running gaits . This passive model, consisting of a point mass and a massless spring leg, can generate a center-of-mass (COM) trajectory and a ground-reaction-force (GRF) profile similar to that of human running . This has encouraged researchers to investigate SLIP to try to understand the fundamental dynamics of bipedal running , and also to use it as a template for controlling real biped robots . For any forward velocity, SLIP can generate periodic running gaits that are passively stable in a narrow region of initial condition parameters, but that are unstable for parameters outside it [5,6]. Therefore, some researchers have proposed stabilizing flight-phase and stance-phase controllers for SLIP running. Flight-phase controllers of the passive SLIP model prepare the swing leg for landing. One bio-inspired strategy uses swing-leg retraction, where the swing leg rotates backwards in the second half of the flight phase [7,8]. A dead-beat controller can reject disturbances by adjusting the attack angle or spring stiffness . A control can update each attack angle to be the negative value of the previous take-off angle . Stance-phase controllers require an actuated SLIP architecture in order to add or remove energy from the system. Schmitt et al  proposed a stabilizing control law for an active SLIP with a force actuator parallel to the spring. Seipel et al  and Ankarali et al  proposed control of an active SLIP with torque actuation at the hip and a spring-damper in the leg. Piovan and Byl  considered an active SLIP model with a displacement actuator in series to the spring and presented a control strategy. (Note that these works add some control actions to the SLIP model in order to stabilize the natural gait).
As an alternative to SLIP, Srinivasan and Ruina  proposed using a point mass biped (PMB) to investigate biped walking and running gaits using numerical optimization. The PMB consists of a point-foot, a point-mass as the body, and a prismatic-force actuator (instead of a spring) on a massless leg. The PMB is “perhaps the simplest mechanical model that is capable of exhibiting a broad range of gaits including walking and running” . They minimized the mechanical-work cost using numerical optimal control methods, obtaining piecewise linear functions that describe optimal walking and running leg-force profiles. They then described finite-force profiles that approximate the impulsive forces. Rebula and Kuo  generated smoother and more human-like force profiles for the same model by introducing an adjustable weighting between work cost and force-like costs (including leg force and its nth time derivatives). They generated single-hump force profiles for optimal running gaits and double-hump profiles for optimal walking gaits. In this work, we also use PMB instead of SLIP, since it is a more general and fundamental biped model which allows one to investigate more general gaits. Because PMB is an active model not restricted to the trajectories and gait specifications imposed by the leg spring of a SLIP model, it provides a better template for real multibody active robots. For example, there is a unique attack angle for each specific forward velocity in SLIP periodic running, but not with a PMB model. Although both SLIP and PMB can generate human-like gaits (qualitatively speaking), in this work we aim to investigate the effects of choosing various analytic stance trajectories on gait optimality. With proper choice of parameters, we find these general analytic trajectories can also generate human-like gait dynamics. Our work is different from the previous works in [15,16], since we can generate any desired trajectory using the same model and then choose the most optimal gait among these general trajectories. The selected general analytic trajectories are also able to generate human-like gait dynamics with proper choice of parameters.
Significant experimental biped robots include the Raibert hopper (a SLIP-like robot with pneumatic actuated legs) , ASIMO (a zero-moment-point controlled walking and running robot with rigid actuation system) , MABEL (a hybrid-zero-dynamics (HZD) walking and running robot with a parallel elastic actuation system) , ATRIAS (a SLIP-based walking and running robot with series elastic actuation system) , and PETMAN (a versatile walking robot developed by Boston Dynamics). However, achieving the agility and efficiency of human running remain open problems. One promising control strategy is to make the real robot follow a minimal model as a template . Poulakakis et al  used SLIP as the HZD of an asymmetric hopper robot in order to generate stable planar hopping motion. Wensing et al  used 3D SLIP as a template to control 3D running and steering of a humanoid robot model. Feedback linearization accomplished tracking in a SLIP COM trajectory of a three-link rigid hopper . Two active SLIP architectures with one and two degrees of actuation were proposed in  as templates for ATRIAS running; the paper proposed a control strategy to follow SLIP, tested successfully in simulations. The results showed that motor torque saturation remains an obstacle to building practical biped running robots.
One aim of our work is to develop techniques that require less peak force for the stance leg. The success of the template-anchor methods described above motivates the approach in this paper where we use a more general and versatile template than the SLIP model. We propose a novel control law to make an underactuated robot’s COM follow any smooth trajectory with any arbitrary initial and final stance conditions (using the PMB model); this result should make it easier to control real biped robots subject to real-world constraints like motor torque saturation. (Note that biped robots with point feet are underactuated in stance phase making them uncontrollable using classic feedback control methods.) Here we compare a circular arc and polynomials of various degrees as COM trajectories in stance phase, but our methods can also be applied to other smooth analytical paths. The variation of cost-of-transport (COT), the maximum GRF with respect to angle of attack, and the COM velocity angle serve as criteria to evaluate the proposed methods.
The benefits of SLIP include its simplicity and its ability to capture some important dynamics of human running . However, it is totally passive. We propose to use PMB in this work in order to capture the advantages of SLIP while avoiding its passivity disadvantage, although we do compare our results to SLIP. PMB consists of a modified SLIP model where a force actuator replaces the spring (Fig 1). The stance leg in both models endures only compressive forces to keep the foot in contact with the ground. Unlike SLIP, PMB can generate arbitrary running gaits.
One step of a running gait includes a stance phase, a take-off, a flight phase and a touch-down. Stance and flight phases are continuous time phases, while take-off and touch-down are instantaneous events. In the stance phase the foot is assumed to be an ideal frictionless passive pivot to the ground. Take-off is an instantaneous transition from stance to flight phase with no discontinuities, occurring when the vertical component of the GRF becomes zero. In flight the robot behaves as a ballistic point mass. Due to the massless legs, there is no impact at touch-down and so no changes occur in the robot’s configuration and velocities. The touch-down state becomes the next stance phase initial condition, with no discontinuities.
In stance phase both SLIP and PMB models have 2 DOFs that can be modeled using Cartesian or polar coordinate systems. The body mass is denoted by m, the leg length by r, the leg angle with respect to vertical by θ, the leg axial force in PMB by Fr, the spring stiffness in SLIP by Ks, the Cartesian unit vectors by i and j, and the polar unit vectors by er and eθ. We assume stance state vector . So constitutes the initial condition of stance. The stance phase equations of motion for SLIP using Cartesian coordinates are (1) which constitutes a passive system that utilizes elastic potential energy to boost the mass. On the other hand, PMB (2) is an active model without elastic elements and uses only control input Fr to boost the mass.
In flight, the ballistic motion of the robot can be described in a straightforward manner using Cartesian coordinates. The massless leg is assumed to be able to move to any desired angle instantaneously with no energy consumption. The flight state vector is . The dynamic model for running can be expressed as in the equation below, where + and - superscripts show the instances just after and just before events, respectively.(3)
Running Gait Generation
For the stance phase trajectories we try a circular arc and polynomials of different degrees. Our method generates running gaits and calculates the necessary leg force profiles in stance phase. The proposed method will work for any desired stance phase path profile consistent with the initial condition.
Circular-arc trajectory in stance phase
We desire a symmetric circular-arc trajectory to be generated using the single force actuator of the leg as shown in Fig 2. The center and the radius of the circle are defined using the touch-down angle, leg length and velocity vector direction. The circle is tangent to the initial velocity vector and its center is located at the intersection of the vertical line passing through the foot contact point and the radius line perpendicular to the initial velocity vector. So the stance phase trajectory becomes symmetric with initial and final positions mirrored with respect to the vertical, and initial and final speeds equal to each other.
Using a polar coordinate system with the origin at foot contact point, the velocity of COM is (4) with r and θ defined in Fig 1. This equation is stated in Cartesian coordinates as (5)
Fig 3 shows the free body diagram of PMB on a circular-arc trajectory. In this figure point O is the foot contact point, point H is the hip point with point mass m located on, and point C is center of the circle. The slope ζ of the line CH at the touch-down moment is perpendicular to the stance initial velocity vector and is calculated as (6)
To calculate the necessary leg force we use a new polar coordinate system with the origin at C, radius of R and angle of θ1 as shown in Fig 3. The unit vectors of this coordinate system are denoted by and . The dynamic equations of PMB in the new coordinate system are (9) where Fr is the leg force and mg is the weight of the robot.
According to Fig 3 the angle θ is (11)
This nonlinear second order differential equation with respect to θ1 is solved numerically with the initial condition of (13) to obtain θ1 and versus time. By substituting this solution into Eq (11) the necessary leg force profile is calculated numerically from Eq (10) to generate the circular-arc trajectory.
Polynomial trajectory in stance phase
As a general framework we define the trajectory of the robot COM to be a polynomial of degree 2n. Since periodic running gaits of SLIP have symmetric stance trajectory, we consider symmetric polynomials with even degrees. The trajectory equation and its time derivatives are written as (14) (15) (16) in which parameters ak can be chosen or designed using COM initial position, velocity and acceleration for stance phase. The Newtonian equations of motion in this phase are written using Eq (2) as (17)
The second order nonlinear differential Eq (18) should be solved numerically with the initial condition of (19) to obtain x, , versus time. Then y versus time is calculated using Eq (14) and substituting those into the first equation of Eq (17) results in the necessary leg force profile to generate the desired polynomial trajectory by COM in stance phase as (20)
Verification of the proposed control law
To verify the proposed control laws to generate arbitrary trajectories for PMB in stance phase, we apply the calculated force profile Fr and the weight to the point mass, with the initial conditions, to generate a trajectory. This trajectory should be identical to the primarily desired circular-arc or polynomial trajectory. To do so, we use equations of motion in polar coordinate system shown in Fig 1b as (21)
Eq (21) can be solved with known Fr and stance initial condition r0, θ0, , to obtain generated trajectories by the proposed control laws in stance phase.
Cost of transport
The cost of transport (COT) is an energy expenditure index that is used to evaluate biped waking and running efficiency. COT is defined as the energy exhausted by the motors per unit weight of the robot per unit distance traveled. To calculate the exhausted energy of motors, we use the absolute value of the product of the motor force and its displacement. (22) where W is the exhausted energy of the motors, mg is the total weight of the robot, L is the range of one running step on the ground, tCS is the time duration of one complete step, Fr is the leg actuator force, and is the leg length variation rate.
A SLIP with a mass of 61.9 kg, a spring stiffness of 16 kN/m and a spring free length of 0.95 m is considered here to generate the running gait. These values are for the equivalent SLIP model for a typical biped robot named ATRIAS . Starting from an appropriate stance phase initial condition, the SLIP running dynamic Eq (3) can be solved to calculate stance and flight trajectories for one step, as well as the initial condition for stance phase of the next step. A SLIP running Poincare map is composed of dynamic equations of one complete step of running and maps the stance phase initial state vector of one step to the initial stance sate vector of the next step. For a given running velocity, a root finding problem can be solved to find the fixed point of this map which shows a stance phase initial condition that generates a periodic running gait. We solved this problem for the above-mentioned SLIP model with touch-down horizontal and vertical velocity components of Vx = 3 m/s and Vy = −0.5 m/s respectively. Its relevant fixed point becomes . Then dynamic Eq (3) are solved with this initial condition that generates a periodic running gait with the desired velocity. Fig 4a shows a SLIP trajectory in stance phase and flight phase. In this figure, thick and thin solid lines show stance and flight trajectories respectively, and dash-dot lines show leg angles in touch-down and take-off events. Fig 4b shows GRF components for this gait in which solid and dashed lines stand for horizontal and vertical components respectively. This figure depicts a single-hump profile for vertical GRF and a sinusoidal shaped profile for horizontal GRF. The COM trajectory and GRF components in Fig 4 are qualitatively similar to human and animal running dynamic specifications .
A PMB model is assumed to have the same mass and free leg length as the SLIP model. Then the proposed gait planning methods are applied to this model to generate various running gaits with circular and polynomial stance trajectories. All gaits have the same initial stance phase as SLIP gait and all have touch-down velocity components of Vx = 3 m/s and Vy = −0.5 m/s. Symmetric trajectory results in a mirrored take-off condition with the state vector of and velocity components of Vx = 3 m/s and Vy = 0.5 m/s. So all generated gaits for PMB will have a fixed step length and fixed touch-down and take-off velocities.
Circular-arc trajectory for PMB in stance phase
Using the desired initial condition we perform the procedure described in section 3.1 step by step to calculate the necessary leg force to generate a circular-arc trajectory in stance phase. The desired trajectory for stance phase is found using L, R and θ1(0) from Eqs (7), (8) and (13). The necessary force profile for the leg actuator is calculated from Eq (9) and then is substituted into the differential equations in Eq (21). These equations are solved with the desired initial condition to calculate the actual trajectory in stance phase. The desired and resulting COM trajectory are exactly the same (Fig 5). This verifies our gait planning method.
The COM trajectory for one complete running step is shown in Fig 6 in which the thick and thin solid lines stand for stance and flight trajectory respectively, and dash-dot lines show leg potion in touch-down and take-off events. It can be seen that the overall trajectory is similar to a SLIP trajectory. The resulting necessary force profile for the leg actuator is depicted in Fig 7 by a thin solid line and its parameters are shown in Table 1. The leg force starts at its maximum value of 1038 N and reaches its minimum value of 872 N in mid-stance. Note it has a different trend than the SLIP force profile which starts at zero and has maximum value of 1463 N in mid-stance. The maximum required leg force for a circular-arc trajectory is 29% less than with SILP, however it starts from its maximum value. Providing such a large force instantaneously just after touch-down is very difficult in practice.
Polynomial trajectories for PMB in stance phase
Polynomials with two parameters.
In this section various polynomial trajectories of form Eq (14) are generated for the stance phase of PMB steady running. The first challenge is how to define the polynomial coefficients using the stance phase initial conditions. The stance phase initial state vector defines the COM initial position [x0, y0]T and velocity . Substituting these points into Eqs (14) and (15) results in two algebraic linear equations with two unknown polynomial coefficients. So we consider even functions with only two coefficients: y = ax2 + b, y = ax4 + b, and y = ax6 + b. These paths with coefficients calculated from the stance phase initial state vector are shown in Fig 8. Using the procedure of Section 3.2 the required leg actuator force profile for each of these trajectories are calculated from Eq (20). Fig 7 and Table 1 show the resulting force profiles. All of them start and end at their maximum value, and reach their minimum value in mid-stance, similar to a circular-arc path. As the number of the degree of the polynomial increases, the maximum value of force grows and its minimum value decreases. The force profile corresponding to a degree-2 polynomial is very close to a circular trajectory force profile. The maximum force increases with polynomial degree. Because practical robots have motor torque restrictions, providing such large forces at the beginning of stance phase is not feasible.
Tuned degree-4 polynomial.
To make the leg force profile start and end at zero, one additional constraint is required in the stance phase initial condition. The acceleration of the body just before touch-down is , where g is the acceleration of gravity. To have the GRF start at zero, this acceleration constraint should be added to the stance phase initial condition. So in this case we have three points (position, velocity and acceleration) to be substituted into Eqs (14), (15) and (16). Therefore three coefficients of the path polynomial are required by these initial conditions, and we choose an even function of degree 4 for the polynomial y = ax4 + bx2 + c. The required leg force to make the robot undergo this trajectory during stance is calculated from Eq (20). Then the resulting force profile is substituted into differential Eq (21) to derive the actual stance trajectory. The desired and actual trajectories are verified to be coincident, similar to Fig 5. Fig 9 shows the PMB trajectory for one step of running using this tuned degree-4 polynomial. This trajectory is very similar to SLIP with the same parameters. The stance phase GRF components for both SLIP and PMB with a tuned degree-4 polynomial are shown in Fig 10. It is observed that these two models have almost the same horizontal GRF components but the vertical component of SLIP is sharper, i.e. the PMB model requires less maximum motor torque and less ground friction coefficient than SLIP. These are the main advantages of PMB with a tuned degree-4 polynomial compared to SLIP running. Note also that physical implementation SLIP-based robots, such as ATRIAS , have motors in series to the spring.
The COM trajectories for one step of running for the all previously generated gaits are shown in Fig 11. All of these gaits have the same touch-down and take-off configuration but different stance trajectories. The flight phases of all these gaits are coincident. In stance phase, it is noticeable that the circular arc and degree-2 polynomial trajectories are very close to each other and the tuned degree-4 polynomial is relatively close to the SLIP trajectory. Horizontal components of COM velocities for these trajectories during stance phase are shown in Fig 7. It is noticeable that again the velocity profiles and the time duration of the circular-arc and degree-2 polynomial trajectories are almost coincident, and the velocity profiles and time duration of SLIP model and degree-4 polynomial trajectories are also close to each other. All of these trajectories start and end with the same velocity of Vx = 3 m/s, but their velocities differ slightly during stance phase. Because the start and end points are fixed, this causes different stance times for these trajectories. The SLIP model has a horizontal velocity of 2.83 m/s in mid-stance and the degree-2 polynomial trajectory has 2.75 m/s. The stance phase duration is 0.200 s for the SLIP model, 0.201 s for the degree-4 trajectory, and 0.205 s for the degree-2 and circular-arc trajectories. All of the trajectories have a flight phase duration of 0.102 s and a complete step stride of 0.888 m. So the average running speed is 2.94 m/s for the SLIP model, 2.93 m/s for the degree-4 trajectory (0.3% less than SLIP), and 2.89 m/s for the degree-2 trajectory (1.7% less than SLIP). Note all of the generated gaits in this paper use the same initial and final position/velocity for stance phase, and different trajectories differ less than 2% in average velocity.
The corresponding leg force profiles for the various trajectories are shown in Fig 12. It is desirable for the leg actuator force in stance phase to have an initial and final value of zero, and a peak value as small as possible. By this criteria, the tuned degree-4 polynomial has the best force profile specifications among these trajectories, with 11% less maximum force than SLIP. The smallest maximum leg force is found with the degree-2 polynomial, and it is 30% less than SLIP. Although this gait needs less motor force, it has the disadvantage of starting and ending at its maximum value. The COTs of the generated gaits are compared in Table 1. Notice that SLIP is a passive model with a COT of zero. In order to gauge whether the PMB gaits are energy efficient, we compare them to a SLIP model where the leg spring has been replaced by a force actuator that needs to generate the same force profile and displacement i.e. a force-actuated SLIP. The resulting force-actuated SLIP has the best COT with a value of 0.2422, but the PMB gaits have very similar COTs (Table 1). The worst PMB COTs are found with the degree-2 polynomial and the circular-arc trajectories, both 1.7% higher than force-actuated SLIP. The tuned degree-4 polynomial PMB has a COT only 1.3% higher.
Optimized degree 6 polynomial.
We conclude that the PMB tuned degree-4 polynomial trajectory meets the main requirements of biped running. In an effort to discover more about optimal gaits, we next use an even function of degree 6 polynomial y = ax6 + bx4 + cx2 + d, where the coefficient a is a free parameter that satisfies the three initial conditions (similar to the tuned degree-4 polynomial). The relevant trajectories with coefficient a from -30 to 10 are shown in Fig 13, which illustrates that the leg length shortens with increasing a. The corresponding leg force profiles are shown in Fig 14, which depicts single-hump profiles for a > −7.22 and double-hump profiles for a < −7.22. (Note the case a = 0 in Fig 14 is identical to the tuned degree-4 polynomial in the previous section). This is an interesting result because a single-hump force profile is known as a characteristics of bipedal running gaits, while a double hump is characteristic of walking [15,16,25]. The double-hump force profile for a running gait is a new result to the best of our knowledge. It is likely that one could capture running gaits with triple and quadruple-hump force profiles by choosing higher-order polynomial trajectories. We claim that a single-hump force profile is not a characteristics of bipedal running gaits (at least in the field of robotics). Although all observed biologic running gaits demonstrate single-hump force profiles, either single or double-hump profiles can be chosen for the running gaits of robots according to engineering requirements. One important restriction for the running gaits of robots is the maximum torque of the motors. To overcome this restriction we need gaits with less maximum leg force. It can be observed in Fig 14 that there is an optimum value for the coefficient a that minimizes maximum leg force, resulting in a double-hump profile. The variation of maximum leg force with respect to a is shown in Fig 15, which shows that the optimum value of the free parameter is a = −12.24. The leg force profile corresponding to the optimal value of coefficient a is shown in Fig 14 by a dash-dot line.
The important specifications of the force profiles of all previously mentioned stance trajectories are summarized in Table 1. The row for the degree-6 polynomial in this table uses the optimal value a = −12.24 and for mid-stance leg force we use the maximum leg force in the double-hump profile. Notice that stance times differ a bit for the various trajectories because we constrained only stance starting velocities, ending velocities, and its path profile (not average running speed). These differences are not significant and do not cause any problems to our investigation.
Effects of Stance Initial Condition on Running Efficiency
We defined the stance phase state vector as . Equivalently the velocity can be defined by Vx, Vy, where the vertical velocity can be written as Vy = Vx tan β according to Fig 16, β is the angle of velocity vector V with respect to the horizontal, and θ is the leg angle with respect to the vertical. So instead we can use [r, θ, Vx, β] as the stance phase state vector and [r0, θ0, Vx0, β0] as the stance initial condition. In this section we examine the effects of attack angle θ0 and velocity angle β0 on the efficiency of the running of a biped robot with free leg length of r0 and touch-down speed of Vx0. Thus r0, Vx0 are assumed to be constant with values of 0.95m and 3m/s respectively.
Because the tuned degree-4 polynomial trajectory includes the necessary biped running dynamics and since we aim to investigate trends of gait specifications, we choose the tuned degree-4 polynomial to generate gaits in this section. For each initial condition [r0, θ0, Vx0, β0] there is a unique tuned degree-4 polynomial. Parameters θ0, β0 are varied from 1° to 34° to generate feasible running gaits. Using the proposed control strategy, the stance trajectory is uniquely defined for each initial condition. The COT and the maximum leg force are calculated in these cases, shown in Figs 17 and 18. It turns out that the COT increases with both θ0 and β0, and it approaches zero when both θ0 and β0 approach zero. Also it can be seen that maximum leg force decreases with θ0 and increases with β0.
Optimal values of stance initial condition within the intervals are summarized in Table 2. The best COT, with value 0.0176, is obtained by the smallest attack angle and velocity angle (θ0 = 1°, β0 = 1°), but this produces a relatively large leg force Fr,max = 1789.8 N. The smallest maximum leg force is obtained by the smallest velocity angle β0 = 1° and the largest attack angle θ0 = 34°, but it results in a COT = 0.346, which is a large value relative to the other gaits shown in Table 1. To find an overall optimum gait considering the trade-off between COT and maximum leg force, we define a normalized index as (23) which is normalized relative to the COT and Fr,max values of the tuned degree-4 polynomial trajectory gait. Minimizing this normalized index generates a gait with θ0 = 4°, β0 = 1 that results in COT = 0.0440 and Fr,max = 1228.2 N. Profiles of leg force components Fx, Fy for the gaits with minimum COT, Fr,max and NI are shown in Fig 19. It is worth noticing that all these gaits have the same horizontal velocity but considerably different time duration and step length. This implies that shorter gaits will need a higher frequency of running steps to travel the same horizontal distance in the same amount of time.
Here we put only the normalized sum of COT and Fr,max as the objective function to obtain an optimal gait. However, humans, animals and robots have some more constraints which should be taken into account when choosing an appropriate gait. For example choosing a relatively small attack angle θ0 = 4° causes an optimal COT and Fr,max, but this requires a high frequency of feet switching that would cause muscle fatigue in humans/animals and challenges in robots due to motor response-time restrictions. Thus choosing a larger attack angle makes more physical sense.
Choosing velocity angle β0 as small as possible is beneficial in minimizing both COT and Fr,max. This is convenient because it has been observed that COM travels on an (almost) horizontal line during high-speed, high-efficiency running in both humans and animals.
A point mass biped (PMB) model provided the basis for generating arbitrary symmetric trajectories in the stance phase of steady running. To show the generality of the method, a circular-arc trajectory and polynomials of various degrees (all satisfying the initial stance conditions) constituted desired trajectories in stance. The algorithm calculated the required force profile for the leg actuator for each trajectory. We found that paths from polynomials with only two parameters (satisfying initial position and velocity) do not start and end with zero force, which is not desirable. To overcome this problem we added the stance phase initial acceleration to the initial condition, then considered paths with three and four parameters. A degree-4 polynomial with three parameters resulted in a path and force profile very similar to those from a SLIP model, with nearly-equal COT and a maximum leg force 11% less. For a degree-6 polynomial with four parameters, optimizing the maximum leg force resulted in a gait with maximum leg force of 25% less than SLIP. The degree-6 polynomial trajectory can generate biped running gaits with either single-hump or double-hump GRF (previously double-hump profiles was counted as a characteristics of walking ). It is worth mentioning that for a defined touch-down velocity, a SLIP model needs a unique attack angle to generate a periodic running gait. However, PMB can generate periodic running gaits with arbitrary touch-down velocities and attack angles (within reasonable limits). We then investigated the effects of attack angle and velocity angle on running efficiency. Choosing the velocity angle as small as possible reduced both COT and maximum leg force. Larger attack angles increased COT and reduced maximum leg force, so there is an optimum value. Using PMB instead of SLIP as a template for multibody biped robots offers some practical advantages: a PMB trajectory can be optimized to overcome real-world limitations such as maximum motor torque, ground coefficient of friction, etc.
We generated symmetric trajectories for the stance phase in this work. Because this was analytic, exact trajectory planning did not deal with stability issues. Implementing these open-loop gaits on consecutive running steps of an underactuated robot will generate periodic unstable gaits. Designing controllers to stabilize these planned underactuated gaits (for example using Poincare map control  or energy level control ) defines future research work. Also, in future work asymmetric trajectories can be investigated in order to reject disturbances in the stance initial condition for steady running, transient running gaits, or running on stairs. Furthermore, this method of gait planning can be applied to 3D PMB and biped robot models with higher degrees of freedom.
The authors would like to thank Dr. Mohammad-Reza S. Noorani and Dr. Behrooz Koohestani for their helpful comments.
- Conceptualization: BD ME CM.
- Data curation: BD ME CM.
- Formal analysis: BD ME CM.
- Funding acquisition: BD ME CM.
- Investigation: BD ME CM.
- Methodology: BD ME CM.
- Project administration: BD CM.
- Resources: BD ME CM.
- Software: BD ME CM.
- Supervision: BD CM.
- Validation: BD ME CM.
- Visualization: BD ME CM.
- Writing – original draft: BD ME CM.
- Writing – review & editing: BD ME CM.
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- 16. Rebula J, Kuo A. The Cost of Leg Forces in Bipedal Locomotion: A Simple Optimization Study. PLOS ONE. 2015; 10(2):e0117384. pmid:25707000
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- 18. Vleugels A. Famous ASIMO robot gets upgraded. United Academics Magazine. November 10, 2011.
- 19. Grizzle JW, Hurst J, Morris B, Park HW and Sreenath K. MABEL, a new robotic bipedal walker and runner. In Proc. Conf. on American Control Conference ACC’09. 2009.
- 20. Grimes J and Hurst J.The design of ATRIAS 1.0 a unique monopod hopping robot. International Conference on Climbing and Walking Robots. Baltimore. USA. 2012.
- 21. Poulakakis I, Grizzle J. The Spring Loaded Inverted Pendulum as the Hybrid Zero Dynamics of an Asymmetric Hopper. IEEE Transactions on Automatic Control. 2009; 54(8):1779–1793.
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How to Calculate the Direct Materials Ending Inventory
Direct materials are the supplies you use to manufacture your inventory: wood for cabinets, steel for girders or wool for coats. Accountants distinguish direct materials from indirect materials, which you use without incorporating them into products. For example, any oil you use to lubricate your factory machinery and any replacement parts you install would be indirect materials. Indirect and direct materials are collectively known as raw materials. When an accounting period ends, you need to calculate your raw materials inventory.
The direct materials inventory for the end of the period equals the beginning inventory, plus purchases, less any direct materials you used up.
Accounting for Direct Materials
Suppose it's time to calculate the direct materials inventory for the previous quarter. You use the same formula for your calculations as you do for figuring the ending inventory of sales stock. Add the beginning materials inventory to the amounts purchased during the quarter. Subtract any materials you used. For example, say you start the quarter with three tons of steel and buy another ton partway through. You use up two tons manufacturing girders. That leaves you with an ending inventory of two tons of direct materials.
If your factory wasted some of the material – a hundredweight of steel scrap, another hundredweight discarded because it was defective, for instance – you still count those amounts as direct materials used in the quarter.
Report on the Balance Sheet
You report the direct materials inventory on the balance sheet for the quarter. The balance sheet combines direct and indirect materials as one entry: raw materials. You add the raw materials to finished goods and work in process for the total inventory. Work in process covers raw materials that have been partially converted to finished goods.
You may discover some of your raw materials have become so degraded that you can't use them. As part of your ending inventory calculation, subtract the value of unusable material from raw materials, and credit it to the cost of goods sold.
Planned Ending Inventory
The ending inventory for one accounting period becomes the starting point for the next period. Planning how much of a direct materials inventory you want on hand protects you against running short and having to stop manufacturing.
Suppose your company sells a line of glass ornaments. Each ornament uses 0.2 pounds of glass. You plan to manufacture 1,000 ornaments in the next quarter, which requires 200 pounds of glass. You started the current quarter with 300 pounds and expect to use 150 pounds. You'll need to buy 50 pounds of glass to have 200 pounds in the ending inventory. That glass will then be ready for use in the next quarter.
Fraser Sherman has written about every aspect of business: how to start one, how to keep one in the black, the best business structure, the details of financial statements. He's also run a couple of small businesses of his own. He lives in Durham NC with his awesome wife and two wonderful dogs.


Part I, Part II, Part III, Part IV, Part V, Part VI, Part VII
In chapter eight, titled "The Road Back Home," Miller discusses some of the implications of the findings of science on a religious person's faith in God. Or, to put it another way, he goes into a theological discussion of what kind of God can be seen as compatible with the findings of modern science. In short, his answer is that the findings of modern science do not at all contradict the traditional God of the western monotheistic religions (Judaism, Christianity, and Islam). This chapter seems to be a catch-all chapter for discussing some of the religious objections to evolution.
First, he lays out three foundational beliefs about God shared by all three western monotheistic religions. These are: 1) the primacy of God in the universe, 2) we exist as the direct result of God's will, and 3) God has revealed himself to us.
Miller then walks the reader through some of the findings of modern science to demonstrate that science does not contradict these core beliefs of theists. He begins with the scientific findings that the universe did in fact have a beginning. If the universe had no beginning, then perhaps the view of God as creator would become untenable. Yet, the universe did have a beginning in the so-called big bang, and science has discovered when it was.
Next, Miller discusses what has been called the anthropic principle. In short, the anthropic principle states that the universe, and specifically a few constants (constant physical forces) in the universe: gravity, strong nuclear force, and electromagnetism, for example, make possible the development of life. If any one of these constants were just slightly higher or lower, then life would not be possible. Miller, quoting Stephen Hawking, writes: "'If the rate of expansion one second after the big bang had been smaller by even one part in a hundred thousand million million it would have recollapsed before it reached its present size.' Conversely, if g were smaller, the dust from the big bang would just have continued to expand, never coalescing into galaxies, stars, planets, or us" (227-228).
Miller notes that these constants by no means prove the existence of God, they also speak to how wonderfully hospitable this universe is to life, such that even strong atheists, such as Daniel Dennet note the danger of the anthropic principle. Miller quotes Dennet as follows:
Believers in any of the proposed strong versions of the Anthropic Principle think they can deduce something wonderful and surprising from the fact the we conscious observers are here--for instance, that in some sense the universe exists for us, or that perhaps we exist so that the universe as a whole can exist, ore even that God created the universe the way He did so that we would be possible (228).The next issue that Miller tackles is the notion of luck or chance. Many religionists object to evolution on the ground that, they say, God would not have left creation to chance, as evolution holds. Yes, Miller concedes, evolution does rely on chance, especially in the appearance of mutations. Yet, for him, the existence of chance in no way nullifies the existence of God. Miller compares evolutionary chance to historical chance. If chance can play a part in history, determining the rise and fall of human empires and the like, then of course it can play a part in evolution. Neither discounts the existence of God. Moreover, without the element of chance, there can be no free will, no free creatures, and no true love of God.
Miller moves on to discussing the role of God in a self-sufficient universe as described by modern science. First of all, Miller does not discount miracles. He claims no philosophical worldview that would deny the possibility of miracles. That said, he says that natural phenomena have natural, scientific explanations. Miller also claims that God can be seen at work through natural, scientific phenomena. For example, he explains that the very existence of the universe cannot be explained through science alone. God, according to Miller is responsible for the existence of the universe.
In short sections, Miller refutes the idea that God would not have taken so long to get to his crowning achievement, humans; and the idea that evolution is too cruel to be the process by which humans were created. His refutation of the first is simply that God is patient and not bound by time as humans are. His answer to the second is twofold. First, cruelty is relative and nature can be seen as cruel, or beautiful. Second, the real surprise in life is not that nature can be cruel, but that something like altruism exists at all.
In the end, I found this the weakest chapter in the book so far. There are some great insights in this chapter, but as a whole, the organization was a little weak and slipshod. Some of Miller's generalizations about religion were not nuanced enough. In the end though, Miller does present plausible reasons why evolution need not be a threat to traditional belief in God.


Until now, all interpretation of the scriptures has made the natural assumption that it was written totally from the viewpoint and imagery of folks who lived in the late Bronze Age. This is largely true but for those parts that were inherited and transcribed into the overall compendium of scriptures. Meanwhile, we have discovered that by the single expedient of changing viewpoint, the chapters on Noah present us with a much more modern and technical interpretation.
I was hesitant to tackle Noah, because I was perhaps unsure if it was relevant to my principal thesis surrounding the Pleistocene Nonconformity. It was not only relevant but an eyewitness accounting of the partial evacuation of Earth before the nonconformity but also a short history of the reestablishment of the human presence from one such effort’s account. It not only fit as a glove, but the intact word usage fitted that of transportation via a magnetic exclusion vessel and clearly had little to do with seamanship. This ended an interpretation problem I have always had with the text, even as a child.
So we are now starting with an entirely different set of assumptions. We presumed the oldest texts are derived from the times of Noah’s settlement establishment around 13000 years ago. These texts were written with the viewpoint of advanced technology and a space based human diaspora supporting the resettlement of Earth with surface prepared humanity after the displacement of the crust. In short, they had the capacity to prepare a set of initial texts that both reflected modern understandings yet was still understandable by a readership lacking those insights.
We have already seen how a description of the Ark morphed into a wooden boat but retained a grasp of size and shape and the ability to lift. This happened in a world in which metal work had shrunk to banging bronze and brittle iron so this is unsurprising.
Let us now walk through Genesis to see if there are any other insights to be suggested. This may also inspire you to see things not seen yet.
1 In the beginning God created the heavens and the earth.
Firstly, what do we mean by earth? The concept of planet does not show up in other creation myths, yet here we find it easily assumed if one does not ask this question. This makes good sense if you understand space and planets. What does it mean if you do not?
This act of creation also accepts my conjecture of particle creation between bounded curvature and unbounded curvature. A snap of the fingers and our universe emerges.
2 Now the earth was [a] formless and empty, darkness was over the surface of the deep, and the Spirit of God was hovering over the waters.
If we substitute unorganized particles for the word ‘earth’ then the first phrase makes sense. An even better reading of the last sentence and this sentence is to substitute the word ‘matter’. Now we have the oddity of the words surface and deep. It is a visual image into which we introduce the Spirit or presence of the presumed creator. We do not introduce a material ‘God’. The oddity of the image of water is introduced as a visual metaphor.
This is pretty good. The idea of creation induced a universe of particles or matter. This conforms exactly to my own conjectures. The presence of God is introduced right at the beginning and described as a spirit independent of matter and is given the task of creation, neatly circumventing of who is first with a circular argument. Welcome to theology.
3 And God said, "Let there be light," and there was light. 4 God saw that the light was good, and He separated the light from the darkness. 5 God called the light "day," and the darkness he called "night." And there was evening, and there was morning—the first day.
Of course, the next step in the evolving universe is the massing of matter leading to the ignition of stars. Naming lit and shadow day and night is fine and also reflects a limited language and little need to explain orbits.
6 And God said, "Let there be an expanse between the waters to separate water from water." 7 So God made the expanse and separated the water under the expanse from the water above it. And it was so. 8 God called the expanse "sky." And there was evening, and there was morning—the second day.
The universe has been created and now exists. We are sitting on a planet. We are describing the sky above and water at our feet and the fact the water also pours from the sky. Thus we call upon God to separate the waters below and above.
9 And God said, "Let the water under the sky be gathered to one place, and let dry ground appear." And it was so. 10 God called the dry ground "land," and the gathered waters he called "seas." And God saw that it was good.
Taking the analogy that one step further we do the obvious and introduce dry land. There is no concept of gravity or a concept of the creation of land as part of the creation process, but it is not particularly necessary.
11 Then God said, "Let the land produce vegetation: seed-bearing plants and trees on the land that bear fruit with seed in it, according to their various kinds." And it was so. 12 The land produced vegetation: plants bearing seed according to their kinds and trees bearing fruit with seed in it according to their kinds. And God saw that it was good. 13 And there was evening, and there was morning—the third day.
Again, an actual explanation of the formation of life is not necessary for this audience. It is enough to know that the land will be covered with plant life. It also does not defy common sense when the reality was just the opposite. By the way there is no reason to think that the order of creation was conserved from the original text. We are reading common sense from the Bronze Age where the option existed to re order the verses to have them fit appearances.
14 And God said, "Let there be lights in the expanse of the sky to separate the day from the night, and let them serve as signs to mark seasons and days and years, 15 and let them be lights in the expanse of the sky to give light on the earth." And it was so. 16 God made two great lights—the greater light to govern the day and the lesser light to govern the night. He also made the stars. 17 God set them in the expanse of the sky to give light on the earth, 18 to govern the day and the night, and to separate light from darkness. And God saw that it was good. 19 And there was evening, and there was morning—the fourth day.
It is time to explain the star filled heavens, so it never needs further discussion. We are sitting on land, surrounded by sky and water and are lit with stars and moon at night. We are almost there. Except why have we not named any stars after Uncle Joe or called them gods? Everyone else did just that. In fact, this reads as a text written by moderns and not by shamans.
In fact, once creation is begun, God acts like a government agent and most likely was in view of our conjecture.
20 And God said, "Let the water teem with living creatures, and let birds fly above the earth across the expanse of the sky." 21 So God created the great creatures of the sea and every living and moving thing with which the water teems, according to their kinds, and every winged bird according to its kind. And God saw that it was good. 22 God blessed them and said, "Be fruitful and increase in number and fill the water in the seas, and let the birds increase on the earth." 23 And there was evening, and there was morning—the fifth day.
We can finally let in all other life forms starting with the seas which is fine, but we also throw in the birds to get them out of the way. This is called poetic license.
24 And God said, "Let the land produce living creatures according to their kinds: livestock, creatures that move along the ground, and wild animals, each according to its kind." And it was so. 25 God made the wild animals according to their kinds, the livestock according to their kinds, and all the creatures that move along the ground according to their kinds. And God saw that it was good.
And on we go to the land animals, including the domesticated stock surrounding them. Again we are slapping paint on the canvass as efficiently as possible.
26 Then God said, "Let us make man in our image, in our likeness, and let them rule over the fish of the sea and the birds of the air, over the livestock, over all the earth, [b] and over all the creatures that move along the ground."
As a last step humanity is created.
27 So God created man in his own image, in the image of God he created him; male and female he created them.
28 God blessed them and said to them, "Be fruitful and increase in number; fill the earth and subdue it. Rule over the fish of the sea and the birds of the air and over every living creature that moves on the ground."
29 Then God said, "I give you every seed-bearing plant on the face of the whole earth and every tree that has fruit with seed in it. They will be yours for food. 30 And to all the beasts of the earth and all the birds of the air and all the creatures that move on the ground—everything that has the breath of life in it—I give every green plant for food." And it was so.
Man is given the explicit task of taming all this. Why did this have to be said at all? Was this not a natural assumption? Yet a group of new settlers could be told this as they began to take charge. God is welcoming them to their new world.
31 God saw all that he had made, and it was very good. And there was evening, and there was morning—the sixth day.
Here we are with a few short verses, with a wonderfully modern explanation of the establishment of man’s place on earth. There are no sub gods or wars and the phraseology would do good service to describe out own understanding of the evolution of life and our cosmos though a little jumbled. God appears as the colonizer in chief and is introduced as a spirit at the beginning to initiate a theological interpretation instead of a real event.


In the campus area, access to the opioid overdose-reversal drug, naloxone, has increased with the passing of House Bill 4, but Ohio State is lagging behind in the effort to make available the life-saving drug.
According to the Centers for Disease Control and Prevention, adults ages 18-25 years old are at the highest risk for heroin addiction. Between 2011 and 2013, heroin use was seen in 7.3 people per 1,000 people, a 109 percent increase from their measure in 2002-2004, which listed 3.5 people per 1,000 people having used heroin.
The bill, which passed last summer, “takes further strides in curtailing Ohio’s opioid epidemic,” according to a House release.
“This legislation will save lives in the State of Ohio, and make this life-saving medication more widely available,” Ohio Rep. Robert Sprague, a primary sponsor of the bill, said in the release.
Pharmacies across the state have already started selling the drug without a prescription, but the Office of Student Life Student Health Services have not yet established the protocols for availability.
“Each pharmacy must create their own protocol for dispensing the medication, and Student Life Student Health Services will develop it as part of an overall medication treatment program for individuals with a diagnosis of substance abuse disorder,” said Dave Isaacs, spokesman for the Office of Student Life, in an emailed statement.
Walgreens and CVS announced in respective press releases in February that both chains had made the drug available over the counter in Ohio.
Ken Hale, associate director of the Higher Education Center for Alcohol and Drug Misuse Prevention and Recovery, said the misuse of prescription opioids, like the painkillers Vicodin and Oxycodone, are directly linked to heroin use. The CDC says that addiction to opioid painkillers makes someone 40 times more likely to use heroin.
“We know that the heroin epidemic we have in the U.S. is growing. House Bill 4 is a great step,” Hale said.
Hale said he hopes to see the university establish a protocol for dispensing the drug without a prescription within a year. For now, if students want to get the drug because they or someone they know has an opioid substance-abuse problem, they can make an appointment at the student health center to get a prescription or go to one of the pharmacies located off campus.
“We have created the perfect storm,” Hale said. “Drugs are part of our culture, we have easy access to medications, and we have lots of misconceptions about using these drugs.”
In an effort to reduce opioid-related deaths and addiction, the CDC’s website offers information about opioids.
The student pharmacy at OSU is having a drug takeback day in April in an effort to get dangerous drugs out of the supply chain, Hale said. Students are encouraged to bring their unfinished prescriptions so that they can be properly disposed.


The making of hand planes for woodworking.
|Status||Critically endangered (see ‘Other information’ for further details)|
|Craft category||Wood; Metal|
|Historic area of significance|
|Area currently practised|
|Origin in the UK||17th century|
|Current no. of professionals (main income)||6-10 (see ‘Other information’ for further details)|
|Current no. of professionals (sideline to main income)
|Current no. of trainees||1|
|Current total no. serious amateur makers
|Current total no. of leisure makers
|Minimum no. of craftspeople required|
Historically, tradesmen would purchase irons from a blacksmith and fashion their own bodies. The first documented professional maker of wooden planes was reportedly Thomas Granford in the late-seventeenth century.
Timber is carefully selected for grain direction and seasoning. It is dimensioned and marked out. The mortise is then chopped out and detailed using traditional plane makers tools, and a wedge is cut and made to fit the mortise. The cutting iron is made from steel, and the then the iron and wedge are fitted to the body so they are a perfect match. Depending on the type of plane there may be a handle or additional fences added. The plane is then tested and chamfered and details.
In terms of making individual or very small batches, wooden planes are typically easier and quicker to produce than infill planes, and as such are priced accordingly. Both can be made either using only hand tools or with a mixture of machine tools and hand tools, but it is more advantageous to use engineering machinery for infill planes because of the resistive nature of metal. Industrial planemaking on a larger scale is carried out by casting the body from iron or bronze and subsequent machining operations before a final assembly of all machined components. Very little handwork is involved in this scale.
Some wooden hand planes are made on machinery by splitting the body down into individual components, forming various geometry that make up the planes function before gluing the body back together into a single piece. The better quality wooden hand planes have their internal geometry morticed out from a single solid block using chisels and specialist plane makers edge tools called ‘floats’. This is superior because their inherent strength is not dependant on glue.
Infill planes are called so because of their construction. A metal shell is formed from thin steel, brass or bronze before being ‘infilled’ with an exotic wood, which supports the blade and forms the ergonomics of the tool. These metal plates are joined to one another via dovetails, which are cut either by hand or more typically on the milling machine. When the two sides are jointed to the sole (bottom) the entire shell is permanently fixed by peening the dovetails, a type of cold forging. The wooden infills are held in place by rivets. This method is superior to casting for longevity because if done correctly, hardly any distortion will occur in the shell, whereas a cast body will invariably warp as it seasons over a century or so.
Blade making and heat treating
Issues affecting the viability of the craft
Business issues: There are lots of issues which affect all small businesses, such as the inability to buy in bulk, and new legislation regarding pensions for all employees.
Market issues: Demand for traditional handmade tools is very limited in the UK.
Market issues: There is definitely a market for high quality hand-made tools. Woodwork is a favoured past-time for people of high net worth with stressful jobs, and there are only half a dozen or so people worldwide making super high end infill style planes.
Market issues: The new, bespoke and high-end handtool market is growing rapidly, particularly in the USA where there are many makers. We have also noticed a decline in the demand for high end vintage/antique hand planes in favour of new ones. This is great but If this trend takes a dive, demand will fall too and some of the makers will fold as a result.
Training issues: There is no structured learning for the craft in any one place. The internet is the best resource but it takes many years of research and practice to properly learn the craft. It depends on what Is being made too on how varied the skill set needs to be – making a handmade wooden plane is completely different from making a handmade infill plane, as is making a cast production plane etc. In reality, every maker is self-trained, or has previous relatable experience that can be utilised.
Market issues: Plane making isn’t very well paid. Bespoke toolmakers agree that they do it for the love of the craft, not the pay check. Customers don’t appreciate the sheer number of hours of incredibly skilled labour that must go into production in order to have a product of that quality, and the investment in materials, machinery and running costs.
Craftspeople currently known
Phil Edwards (Philly planes) – traditional 18th-century wooden-bodied woodworking hand planes, using beech and Sheffield high carbon steel. Full time maker.
Oliver Sparks – custom wood and infill-type planes. Full time maker.
David Barron – Krenov style wooden planes. No longer makers full time, although does some small batches.
Thomas Flinn & Co, Sheffield – there are three skilled craftspeople in the plane department, but this is geared to industrial, factory-style production.
- Theo Cook (Robinson House Studio) – won the 2019 won Future Icons Award for accessories for his collection of hand planes.
Karl Holtey and Bill Carter have recently given up the craft.
Status: 30 years ago plane making was nearing extinction but since then there has been more interest in the craft, helped by the internet and a resurgence in the interest of use for hand tools.
Number of trainees: Thomas Flinn & Co has one trainee, although they are not currently specialising in plane making.
Total number of craftspeople: There are two full-time plane makers plus three plane makers at Thomas Flinn & Co which took over the Clifton brand, although they are geared towards industrial factory production. There are also approximately another five or so hobby makers who are capable of practising the craft but do it occasionally for fun in their spare time rather than professionally, The internet has been great at encouraging hobby makers, and knowledge is gladly shared and published to help people make planes.
- Goodman, W L, (1978) British Planemakers from 1700


US 7883566 B2
A method of using scrap rubber and other scrap materials, such as tires or parts or pieces of tires, to manufacture or melt steel and other metals in a furnace is disclosed. The scrap rubber may be used as a carbon source for the manufacture of steel and other metals, and may be used as an energy source to melt the scrap metal used to make the steel and other metals. The net benefit of this method includes reducing the amount of scrap rubber, such as tires, to be sent to a waste disposal facility or landfill, thereby improving the environment. In addition, by increasing the use of scrap rubber as a source of energy for steel or metal production, less energy is required from other sources.
1. A method of manufacturing steel and other metals or metal alloys comprising the steps of:
applying a quantity of metal into a furnace;
applying energy to the quantity of metal in the furnace to commence the melting of the metal; and
applying a quantity of an energy source to the melting metal in the furnace, wherein the quantity of an energy source does not include added metal,
wherein the quantity of an energy source is used to melt the metal and is selected from the group consisting of whole tires, tire pieces, tire parts, tire sections, tire bales, tire powder, tire particles, tire chunks, waste hoses, carbon-based rubber materials, rubber pieces, rubber parts, rubber sections, rubber powder, rubber particles, rubber chunks, carbon-based waste materials and carbon-based scrap materials.
2. A method of manufacturing steel and other metals or metal alloys comprising the steps of:
applying a quantity of metal into a furnace;
applying energy to the quantity of metal in the furnace to commence the melting of the metal;
transferring the melted metal in a ladle; and
applying a quantity of an energy source to the ladle to further melt the metal or provide a source of carbon, wherein the quantity of an energy source is selected from the group consisting of whole tires, tire pieces, tire parts, tire sections, tire bales, tire powder, tire particles, tire chunks, waste hoses, carbon-based rubber materials, rubber pieces, rubber parts, rubber sections, rubber powder, rubber particles, rubber chunks, carbon-based waste materials, carbon-based pieces, and carbon-based scrap materials.
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14. A method of manufacturing steel and other metals or metal alloys comprising the steps of:
applying a quantity of metal into a furnace; and
separating a quantity of an energy source from a quantity of scrap metal or scrap steel, and
applying the quantity of an energy source to the furnace to further melt the metal or provide a source of carbon, wherein the quantity of an energy source is selected from the group consisting of whole tires, tire pieces, tire parts, tire sections, tire bales, tire powder, tire particles, tire chunks, waste hoses, carbon-based rubber materials, rubber pieces, rubber parts, rubber sections, rubber powder, rubber particles, rubber chunks, carbon-based waste materials, carbon-based pieces, and carbon-based scrap materials.
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This application is a continuation of U.S. application Ser. No. 12/489,837, filed Jun. 23, 2009, now U.S. Pat. No. 7,674,317, which is a continuation of U.S. application Ser. No. 12/165,002, filed Jun. 30, 2008, now U.S. Pat. No. 7,553,351, which is a continuation of U.S. application Ser. No. 11/108,950, filed Apr. 19, 2005, now U.S. Pat. No. 7,393,379, which claims benefit to U.S. Provisional Application Ser. No. 60/563,701, filed Apr. 20, 2004.
The invention relates generally to the melting of scrap metal in a furnace using rubber tires, or parts of rubber tires, as an additional energy source. The invention also relates generally to the inclusion of rubber tires, or parts of rubber tires, as a carbon component for the making of steel and other metals.
It is known to use scrap tires, or parts of tires, as a supplemental fuel source and carbon source for steel melting in an electric arc furnace. The techniques and methods of such use are described in U.S. Pat. No. 5,322,544 and U.S. patent application Ser. No. 09/974,199, both of which are in the name of Franklin Leroy Stebbing and both of which are incorporated herein by reference in their entirety. As described in U.S. Pat. No. 5,322,544, recycling scrap tires into steel or using them as a heat source improves the environment by removing the tires from landfills where they do not degrade but do create a fire hazard. Additionally, a single scrap tire, weighing about 20 pounds, has about the same heating value of coke, approximately 15,000 BTU's per pound, or approximately, 300,000 BTU's per tire.
In the manufacture of steel, scrap tires, which are so plentiful that they have a near zero cost, can be used as a substitute to coal or coke as the heating source, thereby significantly reducing the costs to manufacture the steel. As described in U.S. patent application Ser. No. 09/974,199, the scrap tires can be bundled with scrap steel in a charging bucket and then the contents of the bucket placed in the furnace where the bundled tires and scrap steel are recycled into steel. The present invention described herein builds upon the known techniques and methods of using scrap tires or scrap rubber in the manufacture of steel, as set forth in U.S. Pat. No. 5,322,544 and U.S. patent application Ser. No. 09/974,199.
The present invention provides a system that permits the use of the many forms of tires and scrap rubber in the furnace. The invention allows the tires to be inserted into the furnace at a more precise and controlled rate, time, and location. To this end, for example, by inserting the tires at a slower rate into the furnace and separate from the scrap steel, as compared to batch feeding them or feeding them as tires and scrap steel bundles, it is possible to use larger quantities of tires at a specific controlled location and time. In an exemplary application, four tires or the equivalent of four tires may be added per ton of steel to be produced, as compared to two tires per ton ordinarily expected. A benefit with the increase in the amount of tires that can be used is the decrease in the number and amount of tires to be sent to a waste tire disposal facility or landfill, thereby improving the environment, and putting more scrap tires to beneficial use. In addition, by increasing the use of tires as a source of energy for steel or the metal production, less energy is required from other sources for such steel or metal production.
The invention further allows the use of tires in many forms, so that the tires can be more readily handled and stored before they are put into the furnace. Also, the present invention allows the tires to be put into the furnace separate from the scrap metal that is also to be put into the furnace. Unlike previous methods and techniques, the invention does not require the combining of the tires with the steel or scrap metal prior to the placement of the tires and scrap metal into the furnace. By eliminating the tires from the charge bucket, more available space for the scrap metal is created in the charge bucket. This can be especially valuable in some melt shops where headroom or other factors limit charge bucket size. Also, it is no longer necessary to transport the charging buckets to a location where the tires are stored, as is the case in many electric arc furnace shops.
The invention also allows the tires to be injected into the furnace at very elevated temperatures, thereby increasing the efficiency of the combustion of the tires. Moreover, the invention allows the tires to be injected into the furnace continuously and in a location most advantageous, such as while a foamy slag process is taking place. This allows the carbon monoxide that is being formed by the foamy slag process to convert to carbon dioxide with additional heat generation inside of the furnace—a result of the catalytic effect of the tires. With the invention, the tires may be placed or injected into the molten metal bath in the furnace so as to increase the carbon level in the liquid steel or metal to the desired level for the type and grade being produced.
Other features and advantages of the invention will become apparent to those skilled in the art upon review of the following detailed description, claims and drawings in which like numerals are used to designate like features.
Before the embodiments of the invention are explained in detail, it is to be understood that the invention is not limited in its application to the details of construction and the arrangement of the components set forth in the following description or illustrated in the drawings. The invention is capable of other embodiments and of being practiced or being carried out in various ways. Also, it is to be understood that the phraseology and terminology used herein are for the purpose of description and should not be regarded as limiting. The use of “including” and “comprising” and variations thereof is meant to encompass the items listed thereafter and equivalents thereof as well as additional items and equivalents thereof.
The invention is generally depicted in
To transport to and place the scrap steel or metal into the furnace 10, a number of devices and techniques are possible. The device might be a conventional charging bucket that is filled with scrap steel and is moved to a location above the furnace where the contents can be emptied into the furnace. The device may include a conveyor that is used to convey the scrap steel directly into the furnace in a continuous or semi-continuous mode. A similar device includes a shaft that is filled with scrap metal, and through gravity, the scrap metal moves toward and into the furnace. Typically, the previously mentioned conveyor and shaft have a dual purpose. The conveyor or shaft is enclosed and this enclosure is also used to provide a duct for the removal of the hot exhaust gases from the furnace operation. These hot gases come in contact with the steel or metal scrap to preheat the steel or metal scrap before it enters the furnace.
The aforementioned devices and techniques to transport and place the scrap steel or metal into the furnace 10 may also be used to transport and place the tires, or parts of tires, into the furnace 10 separately from the scrap steel or metal. That is, a conveyor may be used to transport and deposit the tires into the furnace 10, a charging bucket or box may be used to transport and charge the tires into the furnace, or a hopper may be located in proximity to the furnace 10 and used as a feeder to introduce the tires into the furnace 10.
Alternatively, the opening 12 in the furnace 10 roof or sidewall may be used to permit any size tire or tire bales or pieces of tires to be inserted or dropped into the furnace interior and at any desired time and rate so as to maximize the benefits of the melting and refining processes taking place within the furnace 10. The opening 12 can also include a chute to direct the tires into a desired position within the furnace 10. When used with a conveyor fed furnace such as Consteel®, the tires can be dropped into the furnace and the heat from the tires will be drawn through the Consteel® conveyor enclosure by the exhaust fans, thereby providing additional heat to the scrap metal on the conveyor. The tires can be added in a location opposite the conveyor opening to maximize the travel distance from the point of deposit to the conveyor entrance thus maximizing residence time in the furnace for best heat transfer to the scrap in the furnace but also to allow the tires sufficient time to more fully combust before the heat is drawn into the Consteel® enclosure to preheat the scrap. When used with a shaft furnace such as that manufactured by Fuchs, the tires would likewise be deposited to fully maximize the combustion and to gain contact time with the scrap in the furnace. In some instances, it is advantageous to combust the tires alone, prior to adding the scrap steel. This heat energy is drawn through the scrap metal in the shaft or the feeding conveyor, thus preheating it before it is moved into the furnace proper.
As another exemplary alternative, a pneumatic powered gun 14 may be used to put the tires into the furnace 10. In this embodiment, the tires are in a granulated or powdered form or suitable pieces and are blown through a tube or pipe 16 and into the furnace 10. The gun 14 can use compressed air or other gases to propel the tires or tire pieces through the pipe 16. Alternatively, the gun 14 may simply be a tube for inserting the tires and tire pieces into the furnace 10. The gun could be used through the door to “cut in” scrap like a conventional carbon lance and also to provide the added benefit of CO conversion. In a similar fashion, tire pieces can be used to augment or replace carbon in the sidewall burners found in most electric arc furnaces to gain the advantages accruing from the use of tires in place of coal.
The gun 14 can also function as a burner, with the tires or tire pieces serving as the fuel and using air and/or oxygen for combustion. In this embodiment, the tires serve as an additional energy source for the furnace 10. The air can be conveyed by the same pipe as the tires or by a separate pipe 18, as would likely be the case if oxygen were used to promote or enhance the combustion of the tires.
Yet another alternative device or technique for inserting the tires into the furnace 10 includes the use of a thrower device to throw the desired amount of tires into the opening 12 in the furnace 10, or through a door or other suitable opening (not shown). The thrower device would be used as an alternative to the gun in the situations where the tire pieces were not suitable for gunning into the furnace because of the size or shape of the pieces, or if the tires contained wire or bead materials from the tires that were interfering with the operation of the gun. The thrower device may be configured to work in conjunction with the specific size and configuration of tires being used. That is, the thrower device may be configured to accommodate any size tire or combination of tire sizes, along with any size of tire particles, large or small.
By adding the tires separately to the furnace 10, the conventional technique of first combining the metal scrap and the tires, and then putting the combined metal scrap and tires into the furnace is eliminated, thus saving time, effort, and resources. Moreover, this allows the tires to be added separately and directly to the furnace 10 in controlled quantities and at the correct time and location inside of the furnace to obtain maximum benefit. Also more space is created in the charging buckets for additional scrap metal.
More particularly, the objective in operating the furnace is to add energy at the maximum rate possible without causing damage to the furnace, or wasting energy that cannot be absorbed by the scrap metal quickly enough. This is desirable because of the large energy losses that occur to the water cooled furnace walls and roof and also to the furnace fume exhaust duct, combustion chamber, and dust collector. The longer the time taken to bring the scrap to tapping temperature, and/or holding at this temperature, the greater the heat losses. Because of the heat lost through radiation, equipment water cooling, and exhaust gases, the amount of total heat lost is proportional to the time spent in heating the scrap to molten temperature and holding at this temperature before tapping. The higher the rate of heat input, the lower the heat lost to the surroundings because of the time factor. Through the modulated, controlled, insertion of tires to the furnace, separate from the scrap steel, the heat lost is better controlled.
Apart from the heat lost, the cost of energy is also a major consideration, with electricity the most expensive, followed by natural gas, coke, coal and finally scrap tires, which have a negative cost (because they provide a disposal fee for consuming them along with some gain in the scrap steel which they contain). By replacing the traditional heat sources, such as electricity, with scrap tires, significant cost savings can be achieved. More specifically, the cost of electricity is highest at the early stages of the melting cycle because this is the time when the maximum amount of energy can be transferred to the scrap because it is cold and can accept heat more readily and also because the electric arc is shielded by the scrap, reducing damage to the furnace walls and roof. There is a power company “demand charge” for this rapidly consumed electrical energy—a charge that is established by the maximum rate of electrical power used during perhaps a fifteen-minute period in perhaps a twelve-month cycle. Different power companies have differing schedules. With the technique of applying the tires to the furnace separately from the scrap steel, this is also the time when the energy from the scrap tires can be used to reduce the demand charges and replace all or part of the electrical energy consumed during the initial and early stages of the melting cycle, while the scrap is still solid and relatively cold and able to accept the energy from the tire flames as it passes through the scrap. Thus, the use of a modulated, controlled, throttled flow of energy from the tires to replace electricity, (as well as gas, coke or coal) can result in significant cost savings.
Tire energy can be the major source of energy in the early stages of the heating cycle, with natural gas consumption then in a reduced rate and electricity gradually increased as the cycle progress but at a diminished rate at the beginning stages of the heat to save on demand charges. As the scrap metal is turned to molten metal the amount of energy that can be added to the metal bath from sources other than electricity is decreased because the liquid bath surface tends to reflect energy rather than absorb it. Also the bath is protected and covered by the insulating effects of the foamy slag. The solid tire parts can be dropped into or added to the bath, however, increasing carbon content or reacted with oxygen for additional energy in the bath.
The trend in this industry is to reduce electrical energy by adding more chemical energy in the form of natural gas burners and coal because it is cheaper. There are differences in the equipment being used and the melting practices followed, so the results vary from mill to mill. It is clear however, that low cost chemical energy is a very desirable commodity. It should therefore be appreciated that scrap tires clearly supply this need. The rate of energy added to the furnace by electricity, natural gas or coal injection is controlled. The rate of tire energy input needs to be controlled for the same reasons. This further allows the rate of energy release to be reduced so that the tires are not burning at a less controlled rate and consumed before the steel scrap can capture the available heat energy, as can occur if the tires are bundled with the scrap steel prior to insertion into the furnace.
Also, in another aspect of the invention, with the use of tires separate from the scrap steel, the other forms of energy being applied to the furnace may be reduced or regulated in a more precise and controlled manner. As indicated above, these energy inputs are usually natural gas or fuel oil with combustion air or oxygen, electricity, coal or coke. These other energy inputs into the furnace can be more precisely adjusted to correspond to the rate of tire energy input since the tire energy can now be controlled at a desired rate, as opposed to the known techniques of batch feeding the tires with scrap steel. In this invention, the tire energy is the primary source with the other energies filling the gaps not taken care of by tire energy.
Because the tires enhance the conversion of carbon monoxide to carbon dioxide, the tires can be injected into the furnace slag area in a location where this conversion can be most beneficial. The slag having been made from the addition of lime and/or other slag forming materials having the desired flux and foaming properties. The tire injection location is chosen so that the carbon monoxide released from the foamy slag can be converted to carbon dioxide before it enters, or while it is moving through the scrap metal. The carbon monoxide bubbles are formed in the slag by the use of a lance that injects carbon, usually coal particles, along with oxygen gas, directly or indirectly into the slag which is floating on the molten steel puddle in the bottom of the furnace vessel. The foamed slag is in contact with, and partially envelopes and engulfs the steel scrap above it. The carbon monoxide bubbles continually burst and this gas flows through the scrap metal and finally is exhausted out through an exhaust port, usually a hole the furnace roof, often referred to as a “fourth hole.” A large percentage of the available energy in this gas is lost because it exits the furnace as CO rather than converting to CO2. In one method of the invention, the tires or tire pieces would be added to the coal mix during, or prior to, injecting into the slag, or the tires may be added independently of the coal mix. The added tires facilitate the conversion from CO to CO2. The ratio and timing may be adjusted to allow the slag to form CO and then convert to CO2 after the foam had done its job, releasing the conversion energy while it was passing through the scrap metal, thus heating the scrap metal. In another method, the tire or tire pieces would be injected at a location above the slag but low in the scrap layer so there would be sufficient mixing time for the tire fume to react the CO to CO2. While the tires or tire pieces are burning in the scrap metal zone in the furnace, the heat from this exothermic reaction has a chance to contact and transfer to the scrap as it passes through it, thereby improving the rate of melting of the scrap metal. In another method, the tires are used to replace substantially all of the coal for slag foaming. Thus saving the cost of the coal and assisting the CO conversion.
Similarly, the tires can be injected directly into the furnace while steel scrap is being fed by a shaft or conveyor system. The tires are burned in the furnace while the hot gases generated from the tire fire are withdrawn by the fume collection system. This hot exhaust is pulled through the enclosure surrounding the shaft or scrap conveyor, which causes it to be in close contact with the scrap metal. The hot gases will heat the metal as the gases move through the scrap in a counter-flow direction to the direction of travel for the scrap. This greatly enhances the purpose and value for having shaft or conveyor fed furnaces because the amount of available heat captured by them will be much greater. This also increases the efficiency of using tires because the contact time for the flames in these enclosures is even greater because the tires are gradually placed in the furnace so as to burn at a slower, controlled rate, giving the scrap metal more time to absorb the heat. This also allows any remnants of the CO to CO2 conversion to take place in the shaft or conveyor enclosure before it escapes.
By using tires in forms other than whole tires some of the more valuable components of the tires can be removed before the residue from the tires is used for furnace feed. For example, crumb rubber particles have value as raw material for other products. The crumb rubber and other materials can be removed from the tires first, before the residue is used as a carbon, fuel and scrap metal source for steel or metal manufacturing, thus, increasing the value of the scrap tires and helping to convert them from a waste material to valuable assets.
Also, when the rubber material is removed from the tires the remaining tire material has a higher scrap metal content, which can increase its value as a charge material for the furnace. Additionally, having a higher metal content can permit the tire pieces to be handled with a scrap-handling magnet—a device typically found in steel scrap handling facilities and melt shops.
Injecting rubber from the tires directly into a hot furnace, separate from the scrap steel or metal, allows the rubber to be exposed to, and achieve a very high temperature almost instantaneously. This allows the rubber to be incinerated very rapidly with more complete combustion taking place inside of the furnace.
As stated above, the tires can be injected or placed directly into the metal bath so that the carbon from the tire is added directly to the metal. The tires provide a source of carbon, replacing coal or coke. The tires can be injected in controlled amounts so that the level of carbon in the bath can be more precisely adjusted according to the type or grade of steel being produced. This provides a more exact carbon level to be achieved in the finished product. The amount of tires that may be added may vary depending on the level of carbon required for the steel.
In summary, with various aspects of the present invention, the known step of combining the tires together with the metal scrap before introduction into the furnace is eliminated. The efficiency for the use of tires as an energy source is increased, since the energy is consumed gradually, instead of all at once with known batch-feeding techniques, and also the efficient use of the energy from the conversion of CO to CO2 is enhanced because the catalytic effect is not wasted in a rapid release from the furnace before it has time to act. A larger total quantity of tires can be converted to a useful function, greatly improving the environment. Total emissions from the process are improved on a more or less continuous basis. An emission “spike” is eliminated because of the more uniform application of the catalytic effect of the scrap tires. Also, other forms of energy now used in the furnaces can be reduced and replaced by scrap tire energy. With the principles and teachings of the invention, the production of many types of metal products including various steel alloys, iron alloys and the like can be improved.
The above described invention provides several advancements over the art in the use of tires, or parts of tires, as a fuel source for steel or other metal melting, and as a carbon component for the making of steel or other metals. First, the invention improves air emissions from steel melting. Second, the invention provides a way to convert additional waste materials, i.e., scrap tires or tire parts, to a valuable commodity. Third, the invention provides a method for increasing scrap tire consumption by the steel industry. Fourth, the invention eliminates the step of combining the scrap rubber from tires, or tire parts, with scrap metal prior to placing the scrap rubber in the furnace. Fifth, the invention provides a technique of injecting and placing the rubber into the furnace so that the rubber can be put into the furnace in the desired amounts, at the desired times, and in the best location. Sixth, the invention allows the use of the many forms of tires or scrap rubber, e.g., bales, pieces, crumbs, shredded pieces, and the like, as a feed for the furnace. Seventh, the invention allows the reduction of “electrical demand” charges and other electrical consumption charges. Eighth, the invention enhances the scrap preheating aspects of shaft and conveyor fed furnaces, such as Consteel® by increasing the energy for preheating as a result of burning the tires and/or increasing CO to CO2 conversion in the furnace or in the shaft or conveyor. Finally, the invention produces steel or other metals by injecting carbon directly into the liquid metal using scrap rubber as a carbon source for the steel whether the liquid metal is in the furnace or afterward when in the ladle. One skilled in the art will appreciate that the invention provides still other advancements over the art as it pertains to the use of tires, or parts of tires, as a fuel source for steel or other metal melting, and as a carbon component for the making of steel or other metals.
Variations and modifications of the foregoing are within the scope of the present invention. It should be understood that the invention disclosed and defined herein extends to all alternative combinations of two or more of the individual features mentioned or evident from the text and/or drawings. All of these different combinations constitute various alternative aspects of the present invention. The embodiments described herein explain the best modes known for practicing the invention and will enable others skilled in the art to utilize the invention. The claims are to be construed to include alternative embodiments to the extent permitted by the prior art.
Various features of the invention are set forth in the following claims.