Stable bipedal walking with a swing-leg protraction strategy

Transcription

1 To cite this article: Bhounsule, Pranav A., and Zaani, Ali. Stable bipedal walking with a swing-leg protraction strategy. Journal of Bioechanics 51 (017): Stable bipedal walking with a swing-leg protraction strategy Short Counication Pranav A. Bhounsule and Ali Zaani Dept. of Mechanical Engineering, University of Texas San Antonio, One UTSA Circle, San Antonio, TX 7849, USA. Corresponding author: pranav.bhounsule@utsa.edu Abstract In bipedal locootion, swing-leg protraction and retraction refer to the forward and backward otion, respectively, of the swing-leg before touchdown. Past studies have shown that swing-leg retraction strategy can lead to stable walking. We show that swing-leg protraction can also lead to stable walking. We use a siple D odel of passive dynaic walking but with the addition of an actuator between the legs. We use the actuator to do full correction of the disturbance in a single step (a one-step dead-beat control). Specifically, for a given liit cycle we perturb the velocity at id-stance. Then, we deterine the foot placeent strategy that allows the walker to return to the liit cycle in a single step. For a given liit cycle, we find that there is swing-leg protraction at shallow slopes and swing-leg retraction at steep slopes. As the liit cycle speed increases, the swing-leg protraction region increases. On close exaination, we observe that the choice of swing-leg strategy is based on two opposing effects that deterine the tie fro id-stance to touchdown; the walker speed at id-stance and the adjustent in the step length for one-step dead-beat control. When the walker speed doinates, the swing-leg retracts but when the step length doinates, the swing-leg protracts. This result suggests that swing-leg strategy for stable walking depends on the odel paraeters, the terrain, and the stability easure used for control. This novel finding has a clear iplication in the developent of controllers for robots, exoskeletons, and prosthetics and to understand stability in huan gaits. Keywords: Swing-leg Retraction, Walking Stability, Poincaré Map, Dead-beat Control, Locootion 1

2 1 Introduction It has been observed that during huans walking and running, the swing leg oves backward (the angle between the legs is decreasing) prior to touchdown. This is referred to as swing-leg retraction and is hypothesized to stabilize huan gait (Daley and Usherwood, 010; Seyfarth et al., 003; Wisse et al., 005). Model studies of un-actuated achines walking downhill, also known as passive dynaic walking robots (Garcia et al., 1998; McGeer, 1990), have two failies of solutions for certain rap slopes. One solution is stable while the other is unstable. Quite interestingly, the stable solution has a swing-leg retraction while the unstable solution has a swing-leg protraction. This observation led (Wisse et al., 005) to hypothesize that swing-leg retraction helps iprove walking stability. Model studies with controlled swing-leg retraction otions have strengthened this hypothesis. In particular, swing-leg retraction has been deonstrated to increase the stability as easured by the eigenvalue of the Poincaré ap (Wisse et al., 005; Hobbelen and Wisse, 008), and to increase the ability to reject disturbance (e.g., terrain variation) (Hobbelen and Wisse, 008). Swing-leg retraction has also been shown to increase the stability and the robustness to disturbances for odels of running (Blu et al., 010; Karssen et al., 011). In addition, there have been a nuber of odel-based studies that have tried to understand the benefits of swing-leg retraction beyond stabilization. Specifically, swing-leg retraction has been shown to: (1) increase the energy-efficiency by reducing the foot velocity just before touchdown and by reducing the push-off ipulse (Hasaneini et al., 013b,a; Karssen et al., 011); () iniize foot slippage (Karssen et al., 011; Hasaneini et al., 013b); (3) iprove the accuracy of predicting touchdown tiing (Bhounsule et al., 014); (4) decrease peak forces at collisions (Karssen et al., 011); and (5) increase viability and controllability regies (Hasaneini et al., 013b). However, the focus of this paper is on the effect of swing-leg strategy on locootion stability. How does swing-leg retraction iprove walking stability? We provide an explanation based on the paper by (Wisse et al., 005). Consider a D biped odel oving with a certain speed, the noinal speed, which corresponds to a certain energy, the noinal energy, (see Fig. 1A). The noinal values above are evaluated at a particular instance in the walking otion (e.g., at id-stance). Let us assue that the biped speed/energy has changed (e.g., due to a disturbance). In the explanation given below, we are interested in the change in speed, energy, step length, and tie with respect to their noinal values.

3 (1) When the biped is going fast, it has excess energy. This energy can be dissipated by taking a longer step to get back to the noinal speed (see Fig. 1B). This is because an increase in step length (assuing all other factors are held the sae) increases the collisional losses (Ruina et al., 005), thereby eliinating the excess energy. Siilarly, when the biped is going slow, the step length needs to decrease to get back to the noinal speed (see Fig. 1C). Thus changes in the biped speed and step length have a positive correlation. () When the biped is going fast, the step tie will decrease. Siilarly, when the biped is going slow, the step tie will increase. Thus changes in the biped speed and step tie have a negative correlation. Fro (1) and () we see that the step length has a negative correlation with the step tie which iplies a swing-leg retraction strategy (see Fig. 1D). Thus it sees that only swing-leg retraction can lead to stable walking. However, we argue that () is not always true. This is because the step tie depends on; (a) the biped speed, and (b) the step length found in (1). When the biped is going fast, the step length needs to increase to get back to the noinal speed as stated in (1). A faster speed will lead to a decrease in the step tie but the corresponding increase in step length will increase the step tie. When the effect of biped speed doinates the coputation of step tie, there is swing-leg retraction (see Fig. 1D). But when the effect of step length doinates the coputation of step tie, there is swing-leg protraction (see Fig. 1E). Swing-leg protraction can be advantageous because it decreases the energetic cost of swinging the leg. Note that in swing-leg retraction, the hip actuator needs to do ore work to ove the leg beyond its noral step length so that the swing leg has enough tie to be able to ove backward before touchdown. However, in swing-leg protraction the leg is oving forward before touchdown and the actuator does not need to do extra work to ove the leg beyond its noral step length. Thus, the finding that swing-leg protraction helps bipedal stability has iplication for the energy-efficient control of artificial legs in robots, exoskeletons, and prosthetics and in understanding echanics of huan gait. In this paper, we deonstrate that swing-leg protraction can also lead to stable walking and provide an intuitive explanation. Stable walking with swing-leg protraction has also been independently observed by (Safa and Naraghi, 015) in a siilar walking odel but without a hip actuator, but it was ostly ignored as a viable strategy for biped stabilization. Although past results suggest swing-leg retraction leads to stable walking, our results do not contradict the. Specifically, we have exained swing-leg strategies on a broader range of walking otions and terrains not covered in the past studies. Our analysis with a siple bipedal odel suggests that the choice of swing-leg strategy depends on the odel paraeters, the terrain, and the 3

4 stability easure used for control. Thus, further investigation is needed to understand the role of swing-leg otion in gait stabilization. Methods We give brief details of the odel and ethods used to deterine the swing leg control strategy. We use a -diensional odel of walking shown in Fig. siilar to the one used by (Wisse et al., 005). The odel has assless legs of length l and a point-ass M at the hip. Gravity g points downward and the rap slope is γ. The stance leg akes an angle of θ with the vertically downward direction and the swing leg akes an angle of φ with the stance leg. The odel has a rotary hip actuator that can control the swing leg relative to the stance leg. We use a Poincaré ap to relate the state of the walker between successive id-stance positions (see Fig. 3). We define the id-stance to be the position when the gravity vector is along the stance leg. Given the variables at step i, naely, the id-stance velocity ( θ i ), the swing leg angle at touchdown (φ i ), and the rap slope (γ), we can find the id-stance velocity at step i + 1 ( θ i+1 ) using the apping function F as follows: θ i+1 = F ( θ i, φ i, γ). A liit cycle is the steady state otion of the odel. To copute the liit cycle, we put θ i+1 = θ i = θ 0, φ i = φ 0, and γ = γ to get θ 0 = F ( θ 0, φ 0, γ ). (1) See suppleentary aterial for elaborate details on evaluation of F. Stability is defined as the ability of the biped to stick to the sae liit cycle in the presence of a disturbance (e.g., terrain variation, a push). We use the following definition of stability in this analysis: a liit cycle is stable if the biped can fully correct a perturbation in the state in a single step and unstable otherwise. Such a controller that allows for full correction of disturbances in a single step is known as one-step dead-beat control (Antsaklis and Michel, 006). We call this the superstability-based easure. However, the ore widely used stability easure is based on the axiu eigenvalues of F, which we call the eigenvalue-based easure. According to the eigenvalue-based easure, the biped is considered to be stable if the agnitude of the axiu eigenvalue is less than 1, and unstable otherwise (Garcia et al., 1998; Strogatz, 1994). The ain advantage of our easure, the superstability-based easure over the eigenvaluebased easure, is that it is not based on linearization but is ore stringent. Note that both these stability 4

5 easures assess the local stability based on sall perturbations as opposed to global stability, which is the ability not to fall down under sall as well as large perturbations. We state the control proble as follows. Given a id-stance velocity at step i, θ i θ 0, for the given rap slope, γ, we need to find the step length, φ i, required to get back to the noinal id-stance velocity, θ 0, at the next step. Thus θ 0 = F ( θ i, φ i, γ ). () We also need to evaluate the tie fro id-stance to touchdown, t i, which is given as follows t i = θ i 0 0.5φi+γ θ = 0 dθ dθ ( θ i ) + (1 cos θ) (3) Our nuerical calculations are done as follows. To evaluate the liit cycle, we fix the id-stance speed, θ 0, and rap slope, γ, and copute the step length using Eqn. 1. Next, we set θ i = θ 0 in Eqn. 3 to copute the tie fro id-stance to touchdown. Then we vary the id-stance velocity, θ i, and evaluate the swing leg angle, φ i, that will lead to a one-step dead-beat control using Eqn. and the tie fro id-stance to touchdown using Eqn Results and Discussion We show results for two liit cycles in Fig. 4; θ 0 = 0.1 and θ 0 = 0.4. Figs. 4, A and C, show plots of hip angle at touchdown, φ i, vs id-stance velocity, θ i, for various rap slopes. These plots deonstrate that for faster id-stance velocity than the noinal, the walker needs to take a longer step than the noinal to get back to the liit cycle, and vice versa for slower id-stance velocity. The reason is that a longer step length increases while shorter step length decreases the relative energy loss at touchdown (Ruina et al., 005). Figs. 4, B and D, show plots of the hip angle at touchdown, φ i, vs tie to touchdown, t i, as a function of rap slope. These plots give the trajectory that the swing leg should follow to enable a one-step dead-beat control. That is, if the swing leg follows the specific curve for the given rap slope, then the biped is guaranteed to return to the liit cycle on the following step. The gradient at the liit cycle for the plot of swing leg angle versus tie (liit cycle shown by the black 5

6 dot in Fig. 4) deterines the swing-leg strategy; a negative gradient indicates swing-leg retraction while a positive gradient indicates swing-leg protraction. The region of swing-leg protraction is indicated in gray in Fig. 4. Table 1 gives the swing-leg speed for each liit cycle considered in Fig. 4. Fro the table and the figure we note the following: (1) swing-leg protraction at shallow rap slopes and swing-leg retraction at steep rap slopes, and () swing-leg protraction region increases as the liit cycle speed increases, that is, the gray region increases with θ 0. We explain this observation next. The equation for the stance leg is classical inverted pendulu equation and is given by θ = sin θ. We do the sall angle approxiation to rewrite this equation, θ = θ. Next, we solve this equation and use the initial conditions at id-stance, θ(0) = 0 and θ(0) = θ i, to get, θ(t) = θ i sinh(t). At touchdown we have, t = t i and θ(t i ) = 0.5φ i, thus, φ i = θ i sinh t i. (4) We take the differential of the above equation to get φ i = sinh t i ( θ i ) + θ i cosh t i ( t i ). Rearranging the equation, we get the following expression for the swing-leg speed, φ i t i = φi θ i φ i θ i θ i cosh t i sinh t i (5) The sign of the swing-leg speed depends on the ter φi θ i expression are positive. Fro Figs. 4, A and C, we note that φi θ i sinh t i because all the other ters in the at the liit cycle (black dot) is positive but the value decreases as the rap slope increases. That is, at shallow rap slopes, the gradient, φ i θ i, is large, which akes the denoinator in the above equation positive, leading to swing-leg protraction. However, as the rap slope increases, the gradient φi θ, is sall, which akes the denoinator negative, leading to i swing-leg retraction. Further, we note that as the liit cycle speed ( θ ) 0 increases, we get a larger range of rap slopes where the gradient, gray region in D with that in B). φ i θ i, is large, thus increasing the swing-leg protraction region (copare the The transition fro swing-leg retraction to swing-leg protraction occurs through infinity. Fro Figs. 4, B and D, we note that at shallow rap slopes, there is swing leg protraction indicated by positive gradient. As the rap slope increases, the gradient increases, reaching positive infinity. Further increase in the rap slope causes the gradient to flip fro positive to negative infinity and then increases to a finite negative 6

7 value. Thus, for each liit cycle speed, θ, 0 there is rap slope, γ, at which φi θ sinh t i i = 0. When this happens, the swing leg speed is infinite (see Eqn. 5). The physical explanation for this is that the change in leg angle and the change in speed produce an equal and opposite effect on the tie fro id-stance to touchdown. Thus, the tie fro id-stance to touchdown is unchanged leading to an infinite swing leg speed. As infinite speeds are ipossible, the biped loses its ability to be super-stable at this point. The plot of φ i (t i) shown in Figs. 4, B and D can be used to control the hip actuator in exoskeletons and legged robots. When the swing leg is ade to follow the φ i (t i) trajectory, there will be a coplete cancellation of perturbation in speed in a single step, assuing that there are no further disturbances fro id-stance to touchdown. Note that to be able to choose a particular trajectory one requires easureents of id-stance position, id-stance speed, and the rap slope. One interesting question is that: do huans do one-step dead-beat control under perturbations? By the eigenvalue-based stability etric, this would correspond to a axiu eigenvalue of 0. Past studies on huans show that eigenvalues of walking are between 0.4 and 1 (Dingwell and Kang, 007), which suggests an exponential decay rather than dead-beat control. But because huan walking data is noisy and has considerable variability, even if huans did a dead-beat control (in soe diensions), it would not be distinguishable fro an exponential decay. We assued that the swing leg is assless in our odel. But real robots and huans have legs with finite ass. So the question is to whether the results hold true when the legs are assy. The effect of a assy leg is that the swing leg will add/reove energy during the swing phase, in addition to that at touchdown. However, since robots and huans have relatively light legs (legs account for 15% of huan weight (Srinivasan, 006)), we speculate that adding legs to the odel would not alter the results significantly. 4 Conclusions For a siple D point ass odel of walking descending a rap slope there are stable gaits with both, swingleg retraction as well as swing-leg protraction. The reason for different strategies is because the change in tie (copared to noinal step tie) fro id-stance to touchdown depends on two opposing effects; the perturbed id-stance speed and the adjustent in step length. When the speed of walking doinates the coputation of the tie, we obtain swing leg retraction strategy, but when the step length doinates, we obtain swing-leg protraction strategy. For a given liit cycle characterized by a id-stance speed, swing-leg protraction stabilizes walking at shallow rap slopes and swing-leg retraction stabilizes walking at steep 7

8 rap slopes. The swing-leg protraction region increases as the liit cycle speed increases. Our analysis suggests that the swing-leg strategy depends on the rap slope, the noinal walking speed, and the definition of stability, which in our case is full cancellation of perturbations in a single step. Thus it is clear that the swing-leg strategy to stabilize bipedal walking is quite coplex as it depends on a variety of factors. Further research is needed to elucidate the nature of swing-leg strategy for different odels, actuation schees, and stability specification (e.g., eigenvalue-based stability). Conflict of Interest The authors declare that there are no conflicts of interest associated with this work. References Antsaklis, P., Michel, A., 006. Linear Systes, nd ed. Birkhauser, Boston, MA. Bhounsule, P. A., Cortell, J., Grewal, A., Hendriksen, B., Karssen, J. D., Paul, C., Ruina, A., 014. Lowbandwidth reflex-based control for lower power walking: 65 k on a single battery charge. The International Journal of Robotics Research, 33(10), Blu, Y., Lipfert, S., Ruel, J., Seyfarth, A., 010. Swing leg control in huan running. Bioinspiration & Bioietics, 5(), Daley, M. A., Usherwood, J. R., 010. Two explanations for the copliant running paradox: reduced work of bouncing viscera and increased stability in uneven terrain. Biology Letters, 6(3), Dingwell, J. B., Kang, H. G., 007. Differences between local and orbital dynaic stability during huan walking. Journal of Bioechanical Engineering, 19(4), Garcia, M., Chatterjee, A., Ruina, A., Colean, M., The siplest walking odel: Stability, coplexity, and scaling. ASME Journal of Bioechanical Engineering, 10(), Hasaneini, S. J., Macnab, C. J., Bertra, J. E., Leung, H., 013a. Optial relative tiing of stance push-off and swing leg retraction. In Proceedings of 013 IEEE/RSJ International Conference on Intelligent Robots and Systes, Tokyo, Japan. 8

9 Hasaneini, S. J., Macnab, C. J., Bertra, J. E., Ruina, A., 013b. Seven reasons to brake leg swing just before heel strike. In Dynaic Walking, Pittsburgh, PA. Hobbelen, D. G., Wisse, M., 008. Swing-leg retraction for liit cycle walkers iproves disturbance rejection. IEEE Transactions on Robotics, 4(), Karssen, J. D., Haberland, M., Wisse, M., Ki, S., 011. The optial swing-leg retraction rate for running. In Proceedings of 011 IEEE International Conference on Robotics and Autoation, Shanghai, China. McGeer, T., Passive dynaic walking. The International Journal of Robotics Research, 9(), 6 8. Ruina, A., Bertra, J., Srinivasan, M., 005. A collisional odel of the energetic cost of support work qualitatively explains leg sequencing in walking and galloping, pseudo-elastic leg behavior in running and the walk-to-run transition. Journal of Theoretical Biology, 37(), Safa, A. T., Naraghi, M., 015. The role of walking surface in enhancing the stability of the siplest passive dynaic biped. Robotica, 33(1), Seyfarth, A., Geyer, H., Herr, H., 003. Swing-leg retraction: a siple control odel for stable running. Journal of Experiental Biology, 06(15), Srinivasan, M., 006. Why walk and run: energetic costs and energetic optiality in siple echanics-based odels of a bipedal anial. PhD thesis, Cornell University. Strogatz, S., Nonlinear Dynaics and Chaos: With Applications to Physics, Biology, Cheistry, and Engineering. Addison-Wesley, New York, NY. Wisse, M., Atkeson, C., Kloiwieder, D., 005. Swing leg retraction helps biped walking stability. In Proceedings of 005 IEEEE-RAS International Conference on Huanoid Robots, Tsukuba, Japan. 9

10 Mid-stance (step i) At Heel-strike (step i) Mid-stance (step i+1) A Noinal walking cycle θ 0 t 0 t = t = 0 θ 0 φ D Swing Leg Retraction t > > t 0 t 1 Swing Leg not shown φ 0 φ 1 φ 0 B Mid-stance velocity faster than the noinal speed θ 1 > θ 0 t = 0 t = t 1 θ 0 φ t t t 1 0 t φ 1 > φ 0 C Mid-stance velocity slower than the noinal speed θ < θ 0 t = 0 t = t θ 0 E Swing Leg Protraction φ φ 1 φ 0 φ t > > 1 t 0 t φ < φ 0 t t 0 t 1 t Fig. 1: Hypothetical exaple to explain swing leg strategy to enable one-step dead-beat control. (A) Noinal walking cycle. The biped has the sae id-stance leg velocity θ 0 between steps. The noinal step length is φ 0 and tie fro id-stance to touchdown is t 0. (B) The biped starts with id-stance velocity higher than the noinal speed. The biped needs to take a longer than noinal step length φ 1 > φ 0 to increase the collisional loss copared with the noinal gait to get to the noinal id-stance velocity of θ 0. The tie in this case is t = t 1. (C) The biped starts with id-stance velocity lower than the noinal speed. The biped needs to take a shorter than noinal step length φ < φ 0 to reduce the collisional loss copared with the noinal gait to get to the noinal id-stance velocity of θ 0. The tie in this case is, t = t. The swing-leg strategy depends on the tiing t 1 and t relative to t 0 as discussed next. (D) When the ties are such that, t > t 0 > t 1, the swing leg needs to retract in order to regulate walking speed. This is indicated by the negative gradient on the φ vs t plot. The literature has aple exaples of this scenario. (E) When the ties are such that, t 1 > t 0 > t, the swing leg needs to protract in order to regulate the walking speed. This is indicated by the positive gradient on the φ vs t plot.

11 M φ Massless swing leg Swing leg protraction φ > 0 g θ φ < 0 Swing leg retraction γ Massless stance leg Fig. : -D point-ass walking odel. The walker consists of two assless legs of length l with a point-ass M at the hip joint. Gravity points down and is denoted by g. The stance leg (the leg which is on the ground is shown in black) akes an angle θ with the vertically downward direction. The swing leg (the leg which is in the air is shown in light grey) akes an angle φ with the stance leg. We assue that at least one leg is on the ground (single stance phase) and at no instance are both legs on the ground (no double stance phase). The rap slope is γ. There is an actuator at the hip joint.

12 (I) Mid-stance (step i) (II) Before heel-strike (step i) θ i Swing Leg not shown γ θ i - φ i θī (III) After heel-strike (step i) - + φ i + θ i θ i + (IV) Mid-stance (step i+1) θ i +1 Fig. 3: A typical step of our point ass odel. The walker starts in the upright or id-stance position in (I). In this position, the gravity (not shown in the figure) is along the stance leg and in the downward direction. The swing leg is not shown in I. Next, the stance leg (shown in dark color throughout) oves passively under gravity and the swing leg (shown in light gray color throughout) is controlled to follow a tie-based trajectory φ(t). Just before touchdown in (II), the swing leg is at an angle φ i. Next, after touchdown in (III), the swing leg becoes the new stance leg. Finally, the stance leg and the swing leg ove passively. The walker ends in the upright position or id-stance position on the next step in (IV).

13 Liit cycle with id-stance velocity of 0.1 Hip angle at heel-strike, φ i A Slope in degrees Liit cycle (black dot) Mid-stance velocity, θ i Hip angle at heel-strike, φ i B Liit cycle (black dot) Swing leg protratction (indicated by a positive slope) Tie fro id-stance to heel-strike, Slope in degrees t i Hip angle at heel-strike, φ i Liit cycle with id-stance velocity of 0.4 C Slope in degrees Mid-stance velocity, θ i Hip angle at heel-strike, φ i D φ i ( t i ) Tie fro id-stance to heel-strike, 13 Slope in degrees Swing leg protratction (indicated by a positive slope) 15 t 3.5 i Fig. 4: Swing-leg strategy for two liit cycles for a range of rap slopes. Plots for the liit cycle characterized with id-stance velocity θ 0 = 0.1 (A and B) and θ 0 = 0.4 (C and D). The plots on the left colun, A and C, show the hip angle at touchdown (φ i ) vs id-stance velocity ( θ i ) while the plots on the right colun, B and D, show the hip angle at touchdown (φ i ) vs the tie fro id-stance to touchdown (t i). The odel chooses a swing-leg protraction strategy in the grey region and a swing-leg retraction strategy elsewhere.

14 Table 1: Swing-leg speed for the two liit cycles shown in Fig. 4. Slope Swing-leg speed Swing-leg speed for θ 0 = 0.1 for θ 0 =

15 Stable bipedal walking with a swing-leg protraction strategy Suppleentary aterial Pranav A. Bhounsule and Ali Zaani Corresponding author: pranav.bhounsule@utsa.edu, Dept. of Mechanical Engineering, University of Texas San Antonio, One UTSA Circle, San Antonio, TX 7849, USA. 1 Bipedal odel and equations of otion A figure of the odel and a single step of the odel are shown in Fig. and Fig. 3 respectively, in the paper. The step starts in the id-stance phase (stance leg is along the gravitational field) at step i and ends in the id-stance phase at step i + 1. We present the equations of otion next. 1.1 Mid-stance position at step i (Fig. 3, I) to instant before touchdown at step i (Fig. 3, II) The non-diensional id-stance velocity at step i is θ i. We have non-diensionalised the tie with l/g. Fro (I) to (II), the stance leg oves passively to the instant before touchdown, while the swing leg is controlled by the hip actuator to follow a tie-based trajectory φ(t). In (II), the instant before touchdown, the stance leg akes an angle of θ i with the vertical, the swing leg akes an angle of φ with the stance leg, and the non-diensional stance leg velocity is θ i. Since the swing leg is assless, it does not affect the otion of the stance leg. Hence, we ay apply energy conservation fro (I) to (II) to get ( θ i ) + 1 = ( θ i ) + cos θ i. (1) Let non-diensional ground reaction force be R (non-diensionalized with M g). Since the legs are assless, R acts along the stance leg. Using Newton s law we derive an expression for R. Further, the leg 1

16 can only push against the ground. Hence the reaction R needs to be positive. Thus R = cos θ θ 0. () The angular speed, θi, increases onotonically as the angle θ i increases with tie. Thus, we check for the condition given by () only during touchdown (i.e., when the vertical angle is at its axiu after id-stance). Thus cos θ i ( θ i ). (3) Substituting θ i fro Eqn. 1 in Eqn. 3 and siplifying, we get cos θ i ( θ i ) +. (4) 3 Let t i be the tie it takes for the walker to ove fro id-stance position to the instant just before touchdown. Then t i = θ i 0 dθ θ θ = i 0 dθ ( θ i ) + (1 cos θ) (5) which we solve using nuerical quadrature. 1. Instant before touchdown at step i (Fig. 3, II) to instant after touchdown at step i (Fig. 3, III) At touchdown, the legs for a closed loop with the rap. This condition is given by cos(θ i ( φ φ ) i ) cos(θ i ) + sin i sin γ = 0. (6) At touchdown, we switch the angles of the stance and swing legs. To find the angular velocity of the stance leg after touchdown θ + i, we do an angular oentu balance about the ipending collision point

17 (Garcia et al., 1998) θ + i = θ i φ i, (7) φ + i = φ i, (8) θ + i = θ i cos φ i. (9) 1.3 Instant after touchdown at step i (Fig. 3, III) to id-stance position at step i + 1 (Fig. 3, IV) Let the id-stance velocity at step i + 1 be θ i+1. Because the legs are assless, we ay use the conservation of energy to relate the energy of the point ass between III and IV ( θ i+1 ) + 1 = ( θ + i ) + cos θ + i. (10) We also need to ensure that the ground reaction force is positive fro III to IV. Using an arguent siilar to that used to derive Eqn. 3, we get cos θ + i ( θ + i ). (11) Next, we substitute ( θ + i ) fro Eqn. 10 and θ + i fro Eqn. 7 into Eqn. 11 and siplify to get cos(θ i φ i ) ( θ i+1 ) +. (1) 3 Methods To analyze the walking otion we use tools in dynaical systes, naely Poincaré Map and Liit Cycles..1 Poincaré Map The Poincaré ap is used to analyze walking otions of this odel as done by others (Garcia et al., 1998; McGeer, 1990). To copute the ap, we need to relate the state of the walker at any instant in the step with the sae instant on the next step. Here, we will relate the state at id-stance of the current step, i, (see Fig. 3 (I)) to the id-stance at the next step, i + 1 (see Fig. 3 (IV)). To do this, we can use Eqns. 1, 3

18 6, 7, 8, 9, and 10. These 6 equations have 9 variables; θ i, θ i+1, θ + i, θ i, θ+ i, θ i, φ+ i, φ i, and γ. We can use the 6 equations to eliinate 5 variables to end up with an equation with 4 independent variables, θ i+1 = F ( θ i, φ i, γ), (13) where the Poincaré ap F, is a scalar function that aps the state ( θ i ) fro id-stance at step, i, to the state at the next step ( θ i+1 ), i + 1, for a given step length φ i, and rap slope γ.. Liit Cycle Liit cycles are periodic solutions of the Poincaré ap F. To do this, we need to find a fixed point of the function F. Since we want to find solution for a given rap slope, we fix the slope γ = γ in Eqn. 13 and try to find the step length φ i = φ 0 that will lead to θ i+1 = θ i = θ 0. Using the variables above, we can rewrite Eqn. 13 as θ 0 = F ( θ 0, φ 0, γ ). (14).3 One-step dead-beat control A controller that does full correction of disturbances in a single step is known as a one-step dead-beat control (Antsaklis and Michel, 006). We state the one-step dead-beat control as follows. For the disturbance that leads to a id-stance velocity at step i, θ i θ 0, for the given terrain γ, we have to find the step length φ i, needed to get back to the noinal id-stance velocity θ 0 at the next step. Thus θ 0 = F ( θ i, φ i, γ ). (15).4 Nuerical evaluation of liit cycles and dead-beat control Our prie goal is to copute the step length φ i vs tie t i for a given liit cycle. We proceed as follows. Each liit cycle is characterized by a specified id-stance velocity. Thus, the id-stance velocity at the next step is given, θ i+1 = θ 0. The rap slope is given, γ = γ. Next, for a range of id-stance velocities, 0 < θ i < 1, we find values of the 6 unknowns θ + i, θ i, θ+ i, θ i, φ+ i, and φ i using the 6 equations, Eqns. 1, 6, 7, 8, 9, and 10. Also, we rule out solutions that violate the take-off conditions given by Eqns. 4 and 1. We can also copute the tie to go fro id-stance to touchdown, t i, using Eqn. 5 using the coputed 4

19 values of θ i and θ i. Further, we repeat the above calculation for a range of rap slopes, 0.01o < γ < 15 o. Beyond 15 o there are no walking solutions (Bhounsule, 014). References Antsaklis, P., Michel, A., 006. Linear Systes, nd ed. Birkhauser, Boston, MA. Bhounsule, P., 014. Foot placeent in the siplest slope walker reveals a wide range of walking otions. IEEE Transactions on Robotics., 30(5), Garcia, M., Chatterjee, A., Ruina, A., Colean, M., The siplest walking odel: Stability, coplexity, and scaling. ASME Journal of Bioechanical Engineering, 10(), McGeer, T., Passive dynaic walking. The International Journal of Robotics Research, 9(),