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Passively walking five-link robot
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Automatica 40 (2004) 621 – 629
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Brief paper
Passively walking "ve-link robot Elena Borzova, Yildirim Hurmuzlu∗ Mechanical Engineering Department, Southern Methodist University, 5990, Airline Blvd, Dallas, TX 75252, USA Received 17 December 2002; received in revised form 28 August 2003; accepted 27 October 2003
Abstract In this article we investigate the dynamics of a "ve-link, passive bipedal robot. The passivity in this context stands for the ability of the robot to walk autonomously down an inclined surface without any external source of energy. Previous research e6orts in passive walking were limited to four link models with knees or 2-link models without knees with a variety of mass distributions. In this paper we analyze the dynamics of a "ve-link robot with knees and upper body. We were successful in detecting three limit cycles that include three distinct upper body motions. We have investigated the structural stability of these cycles subject to variations in the upper body length. The results demonstrated that the stability can be improved with addition of linear dampers in the hip joints of the model. Also, our investigation demonstrated that erect body posture is only achievable when torsional springs are placed in the hip joints. ? 2003 Published by Elsevier Ltd.
1. Introduction The idea of a biped walking without joint actuation during certain phases of locomotion was initially proposed by Mochon and McMahon (1980a, b). The authors termed this type walk as “Ballistic Walking”. The research in ballistic walking has its origins in the human gait studies that demonstrated a relatively low level of activities in a limb during the swing phase (Basmajian, 1976; Zarrugh, 1976). Subsequently McGeer (1990a) proposed the concept of “Passive Walking”. Passive bipeds walk down a slightly inclined walkway with no external control or energy input. The following models were considered by previous investigators in the "eld: (1) Two element model with a single link stance and swing legs, semicircular feet, and a point mass at the hip (model 1). (2) Four element model with two link stance and swing limbs, circular feet, and a point mass at the hip (model 2). (3) Compass-like passive biped robot (model 3). This paper was not presented at any IFAC Meeting. This paper was recommended for publication in revised form by Associate Editor Jessy W. Grizzle under the direction of Editor Hassan Khalil. ∗ Corresponding author. E-mail address: [email protected] (Y. Hurmuzlu).
0005-1098/$ - see front matter ? 2003 Published by Elsevier Ltd. doi:10.1016/j.automatica.2003.10.015
(4) Point-foot walker with two rigid massless legs hinged at the hip, point-mass at the hip, in"nitesimal point-masses at the feet (model 4). Models 1 and 2 were "rst designed, simulated, and built by McGeer (1990a, 1993). He discovered a stable limit cycle in the motion of both models. McGeer investigated the robustness of the limit cycle with respect to parameter variations. His main observation was that the walking stability was particularly sensitive to model variations in the thigh mass. Goswami, Thuilot, and Espiau (1996) and Thuilot, Goswami and Espiau (1997) studied nonlinear dynamics of a compass-like biped robot (model 3). The model included two variable length members with lumped masses representing the upper body and two limbs. The authors observed limit cycles as well as chaotic trajectories (as a result of successive period doubling bifurcations). They primarily focused on the following parameters: ground slope, mass distribution and limb length. The work of Garcia, Chatterjee, Ruina, and Coleman (1998) and Coleman, Garcia, Ruina, Camp, and Chatterjee (1997) represents a step forward in the research of passively walking bipeds. They used robotic models with rounded and point feet. The authors also considered kneed and straight-legged bipeds (models 1, 2, 4). They showed the existence of walking gaits on arbitrarily small slopes. However, the walking speed also
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y q4 q 3 l3 y
l2
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l1 q*
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approached zero as the slope approached zero. A similar “point-foot walker” biped robot (model 4) was analyzed by Garcia (1998). The robot had two rigid legs connected by a frictionless hinge at the hips. The masses were concentrated at the hip joints and at the feet. Period-doubling bifurcations and chaotic gait were also detected in the motion of the robot model. The main objective of previous research e6orts is to use characteristics of passive gait to design active locomotion systems. The key advantages of passive locomotion include: energy eIciency, self-generated ballistic trajectories, and similarity to natural human gait. Better understanding of the nature of passive walking should allow us to design more eIcient controllers with less stringent torque requirements. Here we present a "ve link passive walking robot, which includes an upper body. The presence of the upper body is important in practical applications. After all, the practical usefulness of the biped without upper body is highly questionable. Yet, as we will see in the paper, the addition of the upper body highly complicates the stability analysis and has a profound e6ect on gait dynamics. In addition, we investigate the e6ect of incorporating passive elements such as springs and dampers in the passive joints of biped. We demonstrate that addition of springs, for example, leads to a new set of gait patterns that do not exist in a motion of a completely passive biped. The stability of the proposed model is studied through the use of Poincare map and Floquet multipliers. 2. The ve element model In this section we present the "ve-link bipedal model shown in Fig. 1. The biped consists of the two identical rigid legs (stance and swing legs) connected by a frictionless hinge at the hip and the rigid upper body. Each leg has a thigh and a shank. When a knee goes into the stance phase
it is locked. The swing knee has a knee-stop preventing hyperextension at the kneestrike. The legs are completed with the semicircular feet. The motion of the robot is constrained to the vertical plane and, therefore, is two dimensional. We consider gaits of the biped that include the single support phase only (i.e. only one of the circular feet is on the ground surface at any given time). The post impact velocity of the contact point of the circular feet with the ground surface is always non zero during the considered gait patterns. We also assume that contacts between the circular feet and the ground are (single) point contacts (see Table 1 in Appendix A for various dimensions, masses, and moments of inertia). The circular feet are designed such that the angle between the stance shank and the line connecting the stance knee to the foot center is q∗ = 0:228 rad (McGeer, 1990b). The motion of the biped includes four stages. The "rst stage is the continuous forward motion during which the circular foot is rolling on the walking surface without slip. We assumed there is suIcient contact friction for this condition. Meanwhile, the other swing leg is moving in the forward direction without any contact with the ground. The second stage begins when the knee strike is detected. This event causes discontinuities in the generalized velocities due to the impact. We assume that the impact is instantaneous and perfectly plastic. The swing knee has a knee-stop preventing hyperextension at the knee strike. In our simulations, the swing knee is locked after the knee strike. The third stage corresponds to the swing phase that follows the knee strike, but before the impact of the swing foot with the ground. The fourth stage begins when the swing foot comes into a sudden contact with the walking surface. The pivot point transfers to the former swing leg and the stance leg is lifted o6. This event causes discontinuities in the generalized positions and velocities. Whereas, the transfer of pivot causes additional discontinuities in the mathematical model due to switching between the swing and the stance sides from bipeds point of view. We assume that the impact is perfectly plastic (i.e. the post impact, normal velocity of the point that contacts the walking surface is zero). We also assume that there is suIcient friction between the feet and the ground surface to prevent slippage. During the continuous phase of the motion the equations of motion are given by the following general form: ˙ q˙ + G(q) = −kq − cq; ˙ M(q)qK + C(q; q)
(1)
where M(q) is the 5 x 5 symmetric, positive de"nite in˙ q˙ is the 5 × 1 vector of centripetal ertia matrix, C(q; q) ˙ is an 5 × 5 matrix), G(q) and coriolis forces (C(q; q) is the 5 × 1 vector representing gravitational forces, k is a diag{k1 ; k2 ; k3 ; k4 ; k5 } spring constant matrix, c is a diag{c1 ; c2 ; c3 ; c4 ; c5 } viscous friction constant matrix. The 5-dimensional vectors q; q˙ and qK represent the joint angles, angular velocities and accelerations, respectively. Equations, representing the impact and switching events can be found in Appendix B.
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3. Analysis of the dynamics The dimensional parameters for our "ve-link model were partially based on the dimensional parameters of the four-link model, introduced by McGeer and then reproduced by Garcia, Chatterjee, and Ruina (1998). The primary driving factor behind such a design was to introduce more dynamic complexity incrementally and, if possible, preserve similarities in dynamic behavior with the well-studied four-link model. For instance, discovering a set of initial conditions that result in stable passive walking is a particularly diIcult problem. Dimensional similarities with the four-link model allowed us to start with the initial conditions previously found for the four-link model by McGeer, Garcia, Chatterjee, and Ruina (1990, 1998). The parameters of the stance shank, the stance knee, the swing shank and the swing knee were set to be exactly equal to the corresponding parameters of the four-link model. To detect the limit cycle of the "ve-link model we perturbed the upper body length parameter and tracked the resulting changes in the respective equilibria of the four link model. We started our simulations with the zero length of the upper body and gradually increased it (Borzova & Hurmuzlu, 2002). Similar approaches were being pursued by others (see Gomes & Ruina, 2003). Phase portraits, Poincare maps, and Floquet multipliers were used to detect the existence of limit cycle and the gait patterns of the "ve-link model. Stability of passive walk is very sensitive to the initial conditions. It may be possible to improve the overall stability of a dynamic system by adding passive elements such as spring and dampers. Such subsystems could serve as additional sources or sinks of energy without violating the overall passivity requirement. The idea of using dampers in a passively walking system was previously explored by Goswami, Thuilot, and Espiau (1998). They analyzed the effect of dampers on the gait of the compass-like passive biped robot. It was shown that inserting the damper at the robot hip joint signi"cantly improved the stability of the compass-like robot gait. Kuo (1999) used spring in his study of the 3-D passive walking machine. It was found that spring is useful for increasing speed of the biped. We decided to extend these ideas by introduction of torsional springs and combinations of both the springs and linear dampers in the hip joints of the model. The main reason for inserting the torsional springs only in the hip joints of the robot was to keep the erect upper body posture of the biped. Meanwhile, the addition of linear dampers in the hip joints of the robot increases the stability of the robot and improves the convergence rate. In this paper we considered the following case studies: Type I: passive gait (k1 = k2 = k3 = k4 = k5 = 0:0; c1 = c2 = c3 = c4 = c5 = 0:0); Type II: passive gait with linear dampers (k1 = k2 = k3 = k4 = k5 = 0:0; c3 = 0:0; c4 = 0:0; c1 = c2 = c5 = 0:0); Type III: passive gait with torsional springs at the hip joins of the robot (k3 = 0:0; k4 = 0:0; k1 =k2 =k5 =0:0; c1 = c2 = c3 = c4 = c5 = 0:0);
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Type IV: passive gait with torsional springs and linear dampers (k3 = 0:0; k4 = 0:0; k1 =k2 =k5 =0:0; c3 = 0:0; c4 = 0:0; c1 = c2 = c5 = 0:0);
4. Type I gait The "rst phase of our study was to detect a limit cycle that existed in the motion of the "ve link model with no springs and dampers at the joints. Fig. 2 depicts the "rst period-one limit cycle that we found. This limit cycle was discovered by taking the four-link model as a base (i.e. zero upper body length). Then we gradually varied the upper body length and modi"ed the original set of initial conditions to obtain the "rst limit cycle depicted in Fig. 2. The speci"c values of the initial conditions were: q1 = 0:423; q˙1 = −1:247; q2 = −0:228; q˙2 = 0:0; q3 = 2:902; q˙3 = −0:081; q4 = −3:487; q˙4 = 1:173; q5 = 0:228; q˙5 = −1:688. As far as parameters are concerned, length of the upper body was set to be equal to 0:063 m, ground slope equal to 0:0495 rad. The phase portrait corresponding to the "rst limit cycle represents "ve hundred locomotion steps, which clearly demonstrates the repeatability of the motion. The four stages of motion that were mentioned above can be marked as follows: − (1) Thl+ − Tkn : Forward motion during which the biped is rolling without slip on the ground on the stance leg. − + (2) Tkn − Tkn : The knee strike or the impact at the knee joint that results in the velocity discontinuity at the knee joint. + (3) Tkn − Thl− : The second stage of the swing phase that comes after the knee strike but before the ground impact. (4) Thl− − Thl+ : The impact of the swing leg with the ground. The pivot point transfers to the former swing leg and the stance leg is lifted o6. This event causes discontinuities in the generalized positions and velocities.
In the present limit cycle the upper body behaves as a pendulum with its center of gravity oscillating below its pivot point. We will further refer to it as Type A walking pattern. This type of motion is quite acceptable from the point of view of robotic biped. Yet it is very important to investigate the existence of gait patterns with erect body posture, since passive gait may also capture the essence of human biped. We will revisit this problem in the latter part of the article. Further experimentation with the initial conditions revealed a second period-one limit cycle. This motion emanates from the following initial conditions: q1 =0:436; q˙1 = −1:24; q2 = −0:228; q˙2 = 0:0; q3 = 3:278; q˙3 = 12:4034; q4 = −3:879; q˙4 = −11:22; q5 = 0:228; q˙5 = −1:827, with identical parameter values used to obtain the "rst limit cycle described above. The circular motion of the upper body is counterclockwise in this case. We will refer to it as Type B
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Fig. 2. Phase portraits for l = 0:063 m, slope angle =0:0495 rad.
walking pattern. The phase portrait for the q1 is also presented in Fig. 2. This phase portrait was constructed for "ve hundred steps. It clearly depicts the existence of the limit cycle. When the initial conditions were slightly varied a third period-four limit cycle appeared in addition to that we have mentioned above. In this case, the upper body is also spinning around the hip, yet in the clockwise direction. We’ll call it Type C walking pattern. This motion emanates from the following initial conditions: q1 =0:433; q˙1 =−1:294; q2 = −0:228; q˙2 =0:0; q3 =−2:458; q˙3 =−13:054; q4 =1:853; q˙4 = 14:218; q5 = 0:228; q˙5 = −1:789. Parameter values are identical to the "rst and second limit cycles. The phase portrait for the q1 is presented in Fig. 2. This phase portrait was constructed for one hundred steps. It clearly depicts the existence of the period-four-limit cycle. The three limit cycles described above represent gait patterns where the erect upper body posture is not maintained. Maintaining the erect upper body posture, however, is one of the most important aspects of bipedal locomotion. In our study we were not able to detect stable Type I gait patterns with erect body posture. 4.1. Stability analysis and bifurcation diagram Poincare map was used as the primary tool for the stability analysis. The Poincare map was "rst applied to the locomotion systems by Hurmuzlu and Moskowitz (1987). The method is based on the construction of a "rst return map by considering the intersection of periodic orbits with an (n − 1)-dimensional cross section in n-dimensional space. In the present study the Poincare section was de"ned as the instant immediately after the heel strike. The map was constructed by using the state values at consecutive steps. The discrete map can be written in the following general form:
i = P( i−1 );
(2)
where is the (n − 1)-dimensional state vector, and the subscripts denote the ith and the i − 1st return value, respectively. Periodic motions of the biped correspond to the "xed points of P
∗ = Pk ( ∗ );
(3)
where Pk is the kth iterate. Stability of Pk reQects the stability of the corresponding Qow. The "xed point ∗ is said to be locally asymptotically stable, if all eigenvalues (Floquet multipliers) of the linearized map, i = DPk ( ∗ ) i−1
(4)
have moduli less than one. The Poincare map can also be utilized to analyze the dynamics of the motion subject to variations in upper body length. The following explanation outlines why the upper body length was chosen as a bifurcation parameter. The initial model had a zero upper body length that is dynamically equivalent to a four-link model. The zero-length model had a stable walking pattern given proper initial conditions. Gradual increase of the upper body length produced other stable walking patterns for a "ve-link model that mainly di6ered in the motion of the upper body, with some stable patterns disappearing and other emerging as the length of the upper body changed. Further analysis showed that the variation of upper body length causes a bifurcation in the case of the Type B walking pattern. Modi"cation of the parameter leads to the change of eigenvalues. At the bifurcation point one of the eigenvalues is equal to −1, which corresponds to the case of the Qip bifurcation (Troger & Steindl, 1991). The bifurcation diagram is presented in Fig. 3. Note that we use the vertical angle (q1 + q2 + q3 ) at the instant of heel strike as the vertical of the axis of the bifurcation diagrams. We chose this variable because this way it is convenient to observe the erect body posture directly from the bifurcation diagram. Due to the period-doubling bifurcation the limit cycle becomes two-periodic. Which results in non-symmetric
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E. Borzova, Y. Hurmuzlu / Automatica 40 (2004) 621 – 629
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Fig. 3. Bifurcation diagrams for slope angle =0:0495 rad.
gait where short step lengths are interlaced with long step lengths. Further increase of the upper body length leads either to the period-eight cycle or to Type A walking pattern (discussed below). Bifurcation diagram for the Type A walking pattern demonstrated that there are no bifurcation points. We were able to increase the length of the upper body by 0:27 m while at the same time maintaining the gait stability. If the length of the upper body is greater than 0:27 m the robot exhibits scuIng (i.e. the swing leg goes underground). The bifurcation diagram is also presented in Fig. 3). Bifurcation diagram for the case corresponding to the Type C walking pattern is also presented in Fig. 3. No bifurcation points were detected for this type of motion. If the length of the upper body is greater than 0:07 m the robot exhibits scuIng (i.e. the swing leg goes underground). 5. Type II gait The addition of dampers leads to a passive gait where the upper body oscillates as a pendulum pivoted at the hip joints (i.e. in the downward direction). This leads to the disappearance to the Type B and Type C walking patterns. Both the convergence rate and the size of the basin of attraction exhibit moderate increases as a result of the added damping. The increase rate of convergence is evidenced by the reduced magnitude of the maximum Floquet multiplier. The larger basin of attraction results in a larger domain for initial conditions that lead to the stable gait. This can be seen on the Poincare map on Fig. 4, which has been constructed for di6erent initial conditions. In order to perform the bifurcation analysis the length of the upper body was chosen as a bifurcation parameter. Fig. 4 depicts the bifurcation diagrams for the three different values of the damping coeIcient. If the length of
the upper body is encreased the limit cycles disappear because of scuIng regardless of the value of the damping coeIcient. 6. Type III gait One of the most important aspects of bipedal locomotion is that the biped should maintain an erect posture during locomotion. For the "ve link passive robot we were not able to detect the set of parameters which would lead to the erect posture during locomotion. The introduction of the dampers improved the stability of the robot, but the direction of the upper body was still downward. The introduction of the torsional springs at the hip joints allowed us to "nd the corresponding set of parameters for the stable gait with erect body posture. The Poincare map is presented in the Fig. 5. The Poincare map clearly shows the existence of the limit cycle. The "gure represents the bifurcation diagrams for di6erent values of spring sti6ness, choosing the upper body length as a bifurcation parameter. We observed the existence of two sets of equilibria in Fig. 5. The "rst set emerges for the higher range of sti6ness, 0:3 6 k 6 2:0 Nm=rad. The range was computed numerically up to one signi"cant digit. As can be observed from the "gure the upper body position for this range of equilibria is almost vertical at the heel strike. The second set of limit cycles takes place at a lower sti6ness range, 0:005 6 k 6 0:14 Nm=rad. The range was computed numerically up to three signi"cant digits. We also note that, for sti6er springs, the upper body lengths that lead to stable limit cycle get longer. For this set of limit cycles, we observe a period doubling bifurcation. We do not analyze the dynamics beyond this bifurcation point, because we are interested only in the period one cycle. (Our main purpose here is to identify limit cycles that will be useful in the control system analysis). Further increase of the upper body
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Fig. 4. Type II gait: (a) Poincare map for l = 0:218 m; c = 0:01 Nm=(rad=sec) slope angle =0:0495 rad; (b) bifurcation diagram for di6erent values of “c”, slope angle =0:0495 rad.
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Fig. 5. Type III gait: (a) Poincare map for l = 0:062 m; k = 0:09 Nm=rad, slope angle =0:0495 rad; (b) bifurcation diagram for k = 0:09 Nm=rad, slope angle =0:0495 rad; (c) bifurcation diagram for di6erent values of “k”, slope angle =0:0495 rad.
length results in the quasi-periodic motion, which also lies outside the scope of this paper. 7. Type IV gait As shown before, the presence of the dampers improved stability characteristic of the gait, while the presence of the springs facilitates keeping the erect body posture during locomotion. Now, we combine springs and dampers to generate stable passive walking with the erect upper body posture. The Poincare map shows the existence of the limit cycle (see Fig. 6). However, the expansion in the basin of attraction was less signi"cant compared to Type II gait. The
rate of convergence to the "xed point has improved. Similar to the Type II gait the Poincare map was constructed for di6erent initial conditions. Fig. 6 presents the bifurcation diagrams for di6erent values of spring sti6ness and damping coeIcient. Similar to Type III gait we observed the existence of two sets of equilibria. For the higher range of sti6ness (0:2 6 k 6 2:0 Nm=rad) and lower damping coeIcient (0:0001 6 c 6 0:001 Nm=(rad=sec) the upper body position is also almost vertical at the heel strike similar to the corresponding case of the spring sti6ness in Type III gait. The introduction of linear dampers allowed us to decrease the vertical angle at the heel strike for k = 0:2 Nm=rad and c = 0:001 Nm=(rad=sec). We could not detect stable gait for k = 0:2 Nm=rad in Type III gait.
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Fig. 6. Type IV gait: (a) Poincare map for k = 0:09 Nm=rad; c = 0:01 Nm=(rad=sec); l = 0:063 m, slope angle 0:0495 rad; (b) bifurcation diagram for di6erent values of “k” and “c”, slope angle 0:0495 rad.
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Fig. 7. Parameters of the passive gait for l = 0:27 m; k = 1:4 Nm=rad; c = 0:004 Nm=(rad=sec): (a) progression velocity v.s. slope angle; (b) vertical angle v.s. slope angle; (c) step length v.s. frequency.
The second set of equilibria corresponds to the following parameters: 0:04 6 k 6 0:3 Nm=rad; 0:01 6 c 6 0:02 Nm= (rad=sec). For this set of limit cycles we did not observe any period doubling bifurcations, as we observed in Type III gait. At the same time it was possible to increase the upper limit of the spring sti6ness range in the presence of the linear dampers from k =0:14 Nm=rad (see Type III gait) to the k = 0:3 Nm=rad, while decreasing the corresponding vertical angle at the heel strike. The lower bound of the spring sti6ness in this case decreased from k =0:005 Nm=rad (see Type III gait) to k = 0:04 Nm=rad. 8. E+ect of the walking surface inclination angle on Type IV gait Another informative parameter variation is the slope angle. The introduction of torsional springs allowed us to maintain the erect upper body posture, while the introduction of
linear dampers increased the convergence rate and the basin of attraction as it was shown above. Our objective here is to study the e6ect of varying the slope angle on the gait of the passive biped with springs and dampers. Introduction of the torsional springs and the linear dampers into the system allowed us to increase the slope by 23% while maintaining gait stability. The deviation of the upper body posture from the vertical line was clearly dependent on the slope angle with the angle of deviation increasing as the slope angle increases (Fig. 7). This walking pattern is similar to the way humans walk down a steep slope. A person typically tries to maintain his balance by deQecting his upper body posture in the direction opposite to that the locomotion. Fig. 7 depicts the variation of the progression velocity with respect to the slope angle. The progression velocity was calculated as the ratio of the step length and the duration of the step in seconds. The slope increase of 23% resulted in the progression velocity increase of 16%.
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The relationship between the step length and the frequency of the step when the slope angle has increased can also provide interesting insights into the nature of the passive gait. The plot is presented in Fig. 7. As seen from the plot when the slope angle increases the biped takes more frequent and longer steps. The main factor in increasing the progression velocity is taking more frequent steps. This walking pattern is also similar to the way humans walk down a steep slope. A person typically tries to maintain his balance by taking more frequent steps.
9. Conclusions In this paper the dynamics of the "ve link model with knees and upper body was analyzed. We were successful in detecting three limit cycles that exhibit three distinct upper body motions. We investigated the structural stability of these cycles subject to variations in the upper body length. We determined that the variation of the upper body length caused the period doubling bifurcation in case of the spinning (counterclockwise) upper body. We demonstrated that the rate of convergence and the stability can be improved with addition of linear dampers in the hip joints of the model. Our investigation also showed that erect body posture is only achievable when torsional springs are placed in the hip joints. In this case the addition of the linear dampers also improved the convergence rate and stability. It was demonstrated that introduction of the torsional springs and the linear dampers into the system allowed us to increase the slope of the walking surface by 23% while maintaining the gait stability. In addition to that, the slope increase of 23% resulted in the progression velocity increase of 23%. The equations of motion were derived using Mathematica package. The numerical simulations were programmed in C++.
Appendix A. The dimensions, masses and moments of inertia are shown in Table 1.
Table 1 Dimensions, masses and moments of inertia Member
Mass Length Mom. of Iner. Center of mass x; y mi (kg) li (m) Ji (kg m2 ) (m)
Stance shank Stance thigh Upper body Swing thigh Swing shank
1.013 2.345 0.223 2.345 1.013
0.44 0.35 0.0589 0.35 0.44
0.039 0.023 0.004 0.023 0.039
x = 0:025; y = 0:1697 x = 0:0; y = 0:09 x = 0:0; y = 0:0589 x = 0:0; y = 0:09 x = 0:025; y = 0:1697
Appendix B. During locomotion two di6erent impacts were considered: knee strike and heel strike. Knee strike was calculated using the principle of generalized momenta. Heel strike leads not only to the discontinuities in the general velocities but there is a sudden exchange in the role of the swing and the stance side members (Hurmuzlu & Moskowitz, 1986; Hurmuzlu, 1993a, b). The following transformation was used to relate the generalized coordinates immediately before and after the impact: 1 1 1 1 1 0 0 0 0 1 − + q ; 0 0 0 1 0 q = (B.1) 0 0 1 0 0 0 1 0 0 0 where the subscript “−” and “+” denote quantities immediately before and after impact and switching, respectively. Impact equations for the heel strike were derived using the principles of linear and angular impulse and momentum. K(q+ ) 0 + R1 (q− ; q˙− ) q˙ −1 0 ; (B.2) ˜ = R2 (q− ; q˙− ) C(q+ ) Fr 0 −1 where q− ; q+ ; q˙− and q˙+ are pre- and post-contact joint positions and velocities, respectively. K is a 5 × 5; C is a 2 × 5; R1 is a 5 × 1 and R2 is a 2 × 1 matrix function.
T F˜ r = F˜ rx ; F˜ ry is the impulse at the contact point of the swing leg with the ground. K(q− )q˙+ represents the angular momentum of the system about the pivot after the impact, R1 (q− ; q˙− ) represents the angular momentum of the system about the same point before impact. C(q+ )q˙+ is the linear momentum of the system after impact and R2 (q+ ; q˙+ ) represents the linear momentum of the system before impact. Eqs. (B.1) and (B.2) were used to calculate the generalized coordinates and generalized velocities immediately after impact in terms of the quantities immediately before impact. For solution to be valid it is necessary to satisfy two additional conditions. The "rst condition is the no slip condition at the contact point. The second condition has to be satis"ed because of the assumption that the circular foot does not experience any impulsive forces immediately after impact: F˜ rx (B.3) ¡ ; F˜ ry y˙ +T ¿ 0;
(B.4)
where is the coeIcient of friction between the contact point and the ground surface and y˙ +T is the velocity of the contact point immediately after the impact.
E. Borzova, Y. Hurmuzlu / Automatica 40 (2004) 621 – 629
References Basmajian, J. (1976). The human bicycle. Biomechanics, 1, 297–302. Borzova, E., & Hurmuzlu, Y. (2002). Passively walking "ve link robot. Proceedings of the sixth biennial conference on engineering systems design and analysis, July 8–11, Istanbul, Turkey. Coleman, M., Garcia, M., Ruina, A., Camp, J., & Chatterjee, A. (1997). Stability and chaos in passive dynamic locomotion. Proceedings of the IUTAM symposium on new applications of nonlinear dynamics and chaos in mechanics, Cornell University, Ithaca, NY. Garcia, M. (1998). Stability, scaling and chaos in passive-dynamic gait models. Ph.D. thesis, Cornell University, Ithaca, NY. Garcia, M., Chatterjee, A., Ruina, A., & Coleman, M. (1998). The simplest walking model: Stability, complexity and scaling. ASME Journal of Biomechanical Engineering, 120(2), 281–288. Gomes, M., & Ruina, A. (2003). A non-singular way to add degrees of freedom: A homotopy method for "nding periodic motions of linked rigid body models. Proceedings of the nineth international symposium on computer simulation in biomechanics, July 2– 4, Sydney, Australia (to appear). Goswami, A., Thuilot, B., & Espiau, B. (1996). Compass-like biped robot, Part 1: Stability and bifurcation of passive gaits. INRIA Research Report, No.2996. Goswami, A., Thuilot, B., & Espiau, B. (1998). A study of the passive gait of a compass-like biped robot: Symmetry and chaos. The International Journal of Robotics Research, 17(12), 1282–1301. Hurmuzlu, Y. (1993a). Dynamics of bipedal gait: Part I—Objective functions and the contact event of a planar "ve-link biped. Journal of Applied Mechanics, 60, 331–336. Hurmuzlu, Y. (1993b). Dynamics of bipedal gait: Part II—Stability analysis of a planar "ve-link biped. Journal of Applied Mechanics, 60, 337–343. Hurmuzlu, Y., & Moskowitz, G. (1986). The role of impact in the stability of bipedal locomotion. Dynamics and Stability of Systems, 1, 217–234. Hurmuzlu, Y., & Moskowitz, G. (1987). Bipedal locomotion stabilized by impact and switching: I. Two and three dimensional, three element models, II. Structural stability analysis of a four element bipedal locomotion model. Dynamics and Stability of Systems, 2, 73–112. Kuo, A. (1999). Stabilization of lateral motion in passive dynamic walking. The International Journal of Robotics Research, 18(9), 917–930.
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McGeer, T. (1990a). Passive dynamic walking. The International Journal of Robotics Research, 9(2), 62–82. McGeer, T. (1990b). Passive walking with knees. Proceedings of the IEEE conference on robotics and automation Vol. 2 (pp. 1640 –1645). McGeer, T. (1993). Dynamics and control of bipedal locomotion. Journal of Theoretical Biology, 163, 277–314. Mochon, S., & McMahon, T. (1980a). Ballistic walking. Journal of Biomechanics, 13, 49–57. Mochon, S., & McMahon, T. (1980b). Ballistic walking: An improved model. Mathematical Biosciences, 52, 241–260. Thuilot, B., Goswami, A., & Espiau, B. (1997). Bifurcation and chaos in a simple passive bipedal gait. IEEE International conference on robotics and automation, New York, Vol. 1 (pp. 792–798). Troger, H., & Steindl, A. (1991). Nonlinear stability and bifurcation theory. Wien: Springer. Zarrugh, M. (1976). Energy and power in human walking. Ph.D. thesis, University of California, Berkley.
Yildirim Hurmuzlu received his Ph.D. degree in Mechanical Engineering from Drexel University. Since 1987, he has been at the Southern Methodist University, Dallas, Texas, where he is a Professor and Chairman of the Department of Mechanical Engineering. His research focuses on nonlinear dynamical systems and Control, with emphasis on robotics, biomechanics, and vibration control. He has published more than 60 articles in these areas. Dr. Hurmuzlu is the associate Editor of the ASME Transactions on Dynamics Systems, Measurement and Control. Elena Borzova was born in Moscow, Russia. She received B.S., M.S. with honors from Moscow State University, Department of Mechanics and Mathematics in 1992. She received Ph.D. in Mechanical Engineering from Southern Methodist University in 2002. Currently she is working as an adjunct professor at Southern Methodist University. Dr. Borzova’s research interests include bipedal locomotion, nonlinear control, system dynamics and bifurcation theory.
Automatica 40 (2004) 621 – 629
www.elsevier.com/locate/automatica
Brief paper
Passively walking "ve-link robot Elena Borzova, Yildirim Hurmuzlu∗ Mechanical Engineering Department, Southern Methodist University, 5990, Airline Blvd, Dallas, TX 75252, USA Received 17 December 2002; received in revised form 28 August 2003; accepted 27 October 2003
Abstract In this article we investigate the dynamics of a "ve-link, passive bipedal robot. The passivity in this context stands for the ability of the robot to walk autonomously down an inclined surface without any external source of energy. Previous research e6orts in passive walking were limited to four link models with knees or 2-link models without knees with a variety of mass distributions. In this paper we analyze the dynamics of a "ve-link robot with knees and upper body. We were successful in detecting three limit cycles that include three distinct upper body motions. We have investigated the structural stability of these cycles subject to variations in the upper body length. The results demonstrated that the stability can be improved with addition of linear dampers in the hip joints of the model. Also, our investigation demonstrated that erect body posture is only achievable when torsional springs are placed in the hip joints. ? 2003 Published by Elsevier Ltd.
1. Introduction The idea of a biped walking without joint actuation during certain phases of locomotion was initially proposed by Mochon and McMahon (1980a, b). The authors termed this type walk as “Ballistic Walking”. The research in ballistic walking has its origins in the human gait studies that demonstrated a relatively low level of activities in a limb during the swing phase (Basmajian, 1976; Zarrugh, 1976). Subsequently McGeer (1990a) proposed the concept of “Passive Walking”. Passive bipeds walk down a slightly inclined walkway with no external control or energy input. The following models were considered by previous investigators in the "eld: (1) Two element model with a single link stance and swing legs, semicircular feet, and a point mass at the hip (model 1). (2) Four element model with two link stance and swing limbs, circular feet, and a point mass at the hip (model 2). (3) Compass-like passive biped robot (model 3). This paper was not presented at any IFAC Meeting. This paper was recommended for publication in revised form by Associate Editor Jessy W. Grizzle under the direction of Editor Hassan Khalil. ∗ Corresponding author. E-mail address: [email protected] (Y. Hurmuzlu).
0005-1098/$ - see front matter ? 2003 Published by Elsevier Ltd. doi:10.1016/j.automatica.2003.10.015
(4) Point-foot walker with two rigid massless legs hinged at the hip, point-mass at the hip, in"nitesimal point-masses at the feet (model 4). Models 1 and 2 were "rst designed, simulated, and built by McGeer (1990a, 1993). He discovered a stable limit cycle in the motion of both models. McGeer investigated the robustness of the limit cycle with respect to parameter variations. His main observation was that the walking stability was particularly sensitive to model variations in the thigh mass. Goswami, Thuilot, and Espiau (1996) and Thuilot, Goswami and Espiau (1997) studied nonlinear dynamics of a compass-like biped robot (model 3). The model included two variable length members with lumped masses representing the upper body and two limbs. The authors observed limit cycles as well as chaotic trajectories (as a result of successive period doubling bifurcations). They primarily focused on the following parameters: ground slope, mass distribution and limb length. The work of Garcia, Chatterjee, Ruina, and Coleman (1998) and Coleman, Garcia, Ruina, Camp, and Chatterjee (1997) represents a step forward in the research of passively walking bipeds. They used robotic models with rounded and point feet. The authors also considered kneed and straight-legged bipeds (models 1, 2, 4). They showed the existence of walking gaits on arbitrarily small slopes. However, the walking speed also
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E. Borzova, Y. Hurmuzlu / Automatica 40 (2004) 621 – 629
y q4 q 3 l3 y
l2
-q2
q1
l1 q*
x y
-q 5
Sl*
x Fig. 1. Planar, "ve element biped.
approached zero as the slope approached zero. A similar “point-foot walker” biped robot (model 4) was analyzed by Garcia (1998). The robot had two rigid legs connected by a frictionless hinge at the hips. The masses were concentrated at the hip joints and at the feet. Period-doubling bifurcations and chaotic gait were also detected in the motion of the robot model. The main objective of previous research e6orts is to use characteristics of passive gait to design active locomotion systems. The key advantages of passive locomotion include: energy eIciency, self-generated ballistic trajectories, and similarity to natural human gait. Better understanding of the nature of passive walking should allow us to design more eIcient controllers with less stringent torque requirements. Here we present a "ve link passive walking robot, which includes an upper body. The presence of the upper body is important in practical applications. After all, the practical usefulness of the biped without upper body is highly questionable. Yet, as we will see in the paper, the addition of the upper body highly complicates the stability analysis and has a profound e6ect on gait dynamics. In addition, we investigate the e6ect of incorporating passive elements such as springs and dampers in the passive joints of biped. We demonstrate that addition of springs, for example, leads to a new set of gait patterns that do not exist in a motion of a completely passive biped. The stability of the proposed model is studied through the use of Poincare map and Floquet multipliers. 2. The ve element model In this section we present the "ve-link bipedal model shown in Fig. 1. The biped consists of the two identical rigid legs (stance and swing legs) connected by a frictionless hinge at the hip and the rigid upper body. Each leg has a thigh and a shank. When a knee goes into the stance phase
it is locked. The swing knee has a knee-stop preventing hyperextension at the kneestrike. The legs are completed with the semicircular feet. The motion of the robot is constrained to the vertical plane and, therefore, is two dimensional. We consider gaits of the biped that include the single support phase only (i.e. only one of the circular feet is on the ground surface at any given time). The post impact velocity of the contact point of the circular feet with the ground surface is always non zero during the considered gait patterns. We also assume that contacts between the circular feet and the ground are (single) point contacts (see Table 1 in Appendix A for various dimensions, masses, and moments of inertia). The circular feet are designed such that the angle between the stance shank and the line connecting the stance knee to the foot center is q∗ = 0:228 rad (McGeer, 1990b). The motion of the biped includes four stages. The "rst stage is the continuous forward motion during which the circular foot is rolling on the walking surface without slip. We assumed there is suIcient contact friction for this condition. Meanwhile, the other swing leg is moving in the forward direction without any contact with the ground. The second stage begins when the knee strike is detected. This event causes discontinuities in the generalized velocities due to the impact. We assume that the impact is instantaneous and perfectly plastic. The swing knee has a knee-stop preventing hyperextension at the knee strike. In our simulations, the swing knee is locked after the knee strike. The third stage corresponds to the swing phase that follows the knee strike, but before the impact of the swing foot with the ground. The fourth stage begins when the swing foot comes into a sudden contact with the walking surface. The pivot point transfers to the former swing leg and the stance leg is lifted o6. This event causes discontinuities in the generalized positions and velocities. Whereas, the transfer of pivot causes additional discontinuities in the mathematical model due to switching between the swing and the stance sides from bipeds point of view. We assume that the impact is perfectly plastic (i.e. the post impact, normal velocity of the point that contacts the walking surface is zero). We also assume that there is suIcient friction between the feet and the ground surface to prevent slippage. During the continuous phase of the motion the equations of motion are given by the following general form: ˙ q˙ + G(q) = −kq − cq; ˙ M(q)qK + C(q; q)
(1)
where M(q) is the 5 x 5 symmetric, positive de"nite in˙ q˙ is the 5 × 1 vector of centripetal ertia matrix, C(q; q) ˙ is an 5 × 5 matrix), G(q) and coriolis forces (C(q; q) is the 5 × 1 vector representing gravitational forces, k is a diag{k1 ; k2 ; k3 ; k4 ; k5 } spring constant matrix, c is a diag{c1 ; c2 ; c3 ; c4 ; c5 } viscous friction constant matrix. The 5-dimensional vectors q; q˙ and qK represent the joint angles, angular velocities and accelerations, respectively. Equations, representing the impact and switching events can be found in Appendix B.
E. Borzova, Y. Hurmuzlu / Automatica 40 (2004) 621 – 629
3. Analysis of the dynamics The dimensional parameters for our "ve-link model were partially based on the dimensional parameters of the four-link model, introduced by McGeer and then reproduced by Garcia, Chatterjee, and Ruina (1998). The primary driving factor behind such a design was to introduce more dynamic complexity incrementally and, if possible, preserve similarities in dynamic behavior with the well-studied four-link model. For instance, discovering a set of initial conditions that result in stable passive walking is a particularly diIcult problem. Dimensional similarities with the four-link model allowed us to start with the initial conditions previously found for the four-link model by McGeer, Garcia, Chatterjee, and Ruina (1990, 1998). The parameters of the stance shank, the stance knee, the swing shank and the swing knee were set to be exactly equal to the corresponding parameters of the four-link model. To detect the limit cycle of the "ve-link model we perturbed the upper body length parameter and tracked the resulting changes in the respective equilibria of the four link model. We started our simulations with the zero length of the upper body and gradually increased it (Borzova & Hurmuzlu, 2002). Similar approaches were being pursued by others (see Gomes & Ruina, 2003). Phase portraits, Poincare maps, and Floquet multipliers were used to detect the existence of limit cycle and the gait patterns of the "ve-link model. Stability of passive walk is very sensitive to the initial conditions. It may be possible to improve the overall stability of a dynamic system by adding passive elements such as spring and dampers. Such subsystems could serve as additional sources or sinks of energy without violating the overall passivity requirement. The idea of using dampers in a passively walking system was previously explored by Goswami, Thuilot, and Espiau (1998). They analyzed the effect of dampers on the gait of the compass-like passive biped robot. It was shown that inserting the damper at the robot hip joint signi"cantly improved the stability of the compass-like robot gait. Kuo (1999) used spring in his study of the 3-D passive walking machine. It was found that spring is useful for increasing speed of the biped. We decided to extend these ideas by introduction of torsional springs and combinations of both the springs and linear dampers in the hip joints of the model. The main reason for inserting the torsional springs only in the hip joints of the robot was to keep the erect upper body posture of the biped. Meanwhile, the addition of linear dampers in the hip joints of the robot increases the stability of the robot and improves the convergence rate. In this paper we considered the following case studies: Type I: passive gait (k1 = k2 = k3 = k4 = k5 = 0:0; c1 = c2 = c3 = c4 = c5 = 0:0); Type II: passive gait with linear dampers (k1 = k2 = k3 = k4 = k5 = 0:0; c3 = 0:0; c4 = 0:0; c1 = c2 = c5 = 0:0); Type III: passive gait with torsional springs at the hip joins of the robot (k3 = 0:0; k4 = 0:0; k1 =k2 =k5 =0:0; c1 = c2 = c3 = c4 = c5 = 0:0);
623
Type IV: passive gait with torsional springs and linear dampers (k3 = 0:0; k4 = 0:0; k1 =k2 =k5 =0:0; c3 = 0:0; c4 = 0:0; c1 = c2 = c5 = 0:0);
4. Type I gait The "rst phase of our study was to detect a limit cycle that existed in the motion of the "ve link model with no springs and dampers at the joints. Fig. 2 depicts the "rst period-one limit cycle that we found. This limit cycle was discovered by taking the four-link model as a base (i.e. zero upper body length). Then we gradually varied the upper body length and modi"ed the original set of initial conditions to obtain the "rst limit cycle depicted in Fig. 2. The speci"c values of the initial conditions were: q1 = 0:423; q˙1 = −1:247; q2 = −0:228; q˙2 = 0:0; q3 = 2:902; q˙3 = −0:081; q4 = −3:487; q˙4 = 1:173; q5 = 0:228; q˙5 = −1:688. As far as parameters are concerned, length of the upper body was set to be equal to 0:063 m, ground slope equal to 0:0495 rad. The phase portrait corresponding to the "rst limit cycle represents "ve hundred locomotion steps, which clearly demonstrates the repeatability of the motion. The four stages of motion that were mentioned above can be marked as follows: − (1) Thl+ − Tkn : Forward motion during which the biped is rolling without slip on the ground on the stance leg. − + (2) Tkn − Tkn : The knee strike or the impact at the knee joint that results in the velocity discontinuity at the knee joint. + (3) Tkn − Thl− : The second stage of the swing phase that comes after the knee strike but before the ground impact. (4) Thl− − Thl+ : The impact of the swing leg with the ground. The pivot point transfers to the former swing leg and the stance leg is lifted o6. This event causes discontinuities in the generalized positions and velocities.
In the present limit cycle the upper body behaves as a pendulum with its center of gravity oscillating below its pivot point. We will further refer to it as Type A walking pattern. This type of motion is quite acceptable from the point of view of robotic biped. Yet it is very important to investigate the existence of gait patterns with erect body posture, since passive gait may also capture the essence of human biped. We will revisit this problem in the latter part of the article. Further experimentation with the initial conditions revealed a second period-one limit cycle. This motion emanates from the following initial conditions: q1 =0:436; q˙1 = −1:24; q2 = −0:228; q˙2 = 0:0; q3 = 3:278; q˙3 = 12:4034; q4 = −3:879; q˙4 = −11:22; q5 = 0:228; q˙5 = −1:827, with identical parameter values used to obtain the "rst limit cycle described above. The circular motion of the upper body is counterclockwise in this case. We will refer to it as Type B
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E. Borzova, Y. Hurmuzlu / Automatica 40 (2004) 621 – 629
Type A
Tkn
-0.8 -1.0
−
−
Tkn
Thl
+ hl
T
-1.4 -0.1
0.0
+
Tkn
-0.8
.
-1.2
Type B -0.6
+
q1 (rad/s)
.
q1 (rad/s)
-0.6
0.1
0.2 q1 (rad)
0.3
-1.0 -1.2 -1.4
0.4
−
−
Tkn
Thl
-0.1
0.0
0.1
0.2 q1 (rad)
Thl+ 0.3
0.4
Type C +
.
q1 (rad/s)
-0.6
Tkn
-0.8 -1.0 -1.2
−
−
Thl
Tkn Thl+
-1.4 -0.1
0.0
0.1
0.2 q1 (rad)
0.3
0.4
Fig. 2. Phase portraits for l = 0:063 m, slope angle =0:0495 rad.
walking pattern. The phase portrait for the q1 is also presented in Fig. 2. This phase portrait was constructed for "ve hundred steps. It clearly depicts the existence of the limit cycle. When the initial conditions were slightly varied a third period-four limit cycle appeared in addition to that we have mentioned above. In this case, the upper body is also spinning around the hip, yet in the clockwise direction. We’ll call it Type C walking pattern. This motion emanates from the following initial conditions: q1 =0:433; q˙1 =−1:294; q2 = −0:228; q˙2 =0:0; q3 =−2:458; q˙3 =−13:054; q4 =1:853; q˙4 = 14:218; q5 = 0:228; q˙5 = −1:789. Parameter values are identical to the "rst and second limit cycles. The phase portrait for the q1 is presented in Fig. 2. This phase portrait was constructed for one hundred steps. It clearly depicts the existence of the period-four-limit cycle. The three limit cycles described above represent gait patterns where the erect upper body posture is not maintained. Maintaining the erect upper body posture, however, is one of the most important aspects of bipedal locomotion. In our study we were not able to detect stable Type I gait patterns with erect body posture. 4.1. Stability analysis and bifurcation diagram Poincare map was used as the primary tool for the stability analysis. The Poincare map was "rst applied to the locomotion systems by Hurmuzlu and Moskowitz (1987). The method is based on the construction of a "rst return map by considering the intersection of periodic orbits with an (n − 1)-dimensional cross section in n-dimensional space. In the present study the Poincare section was de"ned as the instant immediately after the heel strike. The map was constructed by using the state values at consecutive steps. The discrete map can be written in the following general form:
i = P( i−1 );
(2)
where is the (n − 1)-dimensional state vector, and the subscripts denote the ith and the i − 1st return value, respectively. Periodic motions of the biped correspond to the "xed points of P
∗ = Pk ( ∗ );
(3)
where Pk is the kth iterate. Stability of Pk reQects the stability of the corresponding Qow. The "xed point ∗ is said to be locally asymptotically stable, if all eigenvalues (Floquet multipliers) of the linearized map, i = DPk ( ∗ ) i−1
(4)
have moduli less than one. The Poincare map can also be utilized to analyze the dynamics of the motion subject to variations in upper body length. The following explanation outlines why the upper body length was chosen as a bifurcation parameter. The initial model had a zero upper body length that is dynamically equivalent to a four-link model. The zero-length model had a stable walking pattern given proper initial conditions. Gradual increase of the upper body length produced other stable walking patterns for a "ve-link model that mainly di6ered in the motion of the upper body, with some stable patterns disappearing and other emerging as the length of the upper body changed. Further analysis showed that the variation of upper body length causes a bifurcation in the case of the Type B walking pattern. Modi"cation of the parameter leads to the change of eigenvalues. At the bifurcation point one of the eigenvalues is equal to −1, which corresponds to the case of the Qip bifurcation (Troger & Steindl, 1991). The bifurcation diagram is presented in Fig. 3. Note that we use the vertical angle (q1 + q2 + q3 ) at the instant of heel strike as the vertical of the axis of the bifurcation diagrams. We chose this variable because this way it is convenient to observe the erect body posture directly from the bifurcation diagram. Due to the period-doubling bifurcation the limit cycle becomes two-periodic. Which results in non-symmetric
Type A
3.16
vertical angle (rad)
vertical angle (rad)
E. Borzova, Y. Hurmuzlu / Automatica 40 (2004) 621 – 629
3.14 3.12 3.10 3.08 0.05
0.10
0.15
0.20
Type B bifurcation point
-2.6 -2.8 -3.0
0.25
30
length of the upper body (m)
Type C
40 50 60x10 length of the upper body (m)
-3
3
vertical angle (rad)
vertical angle (rad)
3
625
2 1 0 -1 -2 -3
Three limit cycles
2 1 0 -1 -2 -3
0.00
0.02
0.04
0.06
0.08
length of the upper body (m)
0.05
0.10
0.15
0.20
0.25
length of the upper body (m)
Fig. 3. Bifurcation diagrams for slope angle =0:0495 rad.
gait where short step lengths are interlaced with long step lengths. Further increase of the upper body length leads either to the period-eight cycle or to Type A walking pattern (discussed below). Bifurcation diagram for the Type A walking pattern demonstrated that there are no bifurcation points. We were able to increase the length of the upper body by 0:27 m while at the same time maintaining the gait stability. If the length of the upper body is greater than 0:27 m the robot exhibits scuIng (i.e. the swing leg goes underground). The bifurcation diagram is also presented in Fig. 3). Bifurcation diagram for the case corresponding to the Type C walking pattern is also presented in Fig. 3. No bifurcation points were detected for this type of motion. If the length of the upper body is greater than 0:07 m the robot exhibits scuIng (i.e. the swing leg goes underground). 5. Type II gait The addition of dampers leads to a passive gait where the upper body oscillates as a pendulum pivoted at the hip joints (i.e. in the downward direction). This leads to the disappearance to the Type B and Type C walking patterns. Both the convergence rate and the size of the basin of attraction exhibit moderate increases as a result of the added damping. The increase rate of convergence is evidenced by the reduced magnitude of the maximum Floquet multiplier. The larger basin of attraction results in a larger domain for initial conditions that lead to the stable gait. This can be seen on the Poincare map on Fig. 4, which has been constructed for di6erent initial conditions. In order to perform the bifurcation analysis the length of the upper body was chosen as a bifurcation parameter. Fig. 4 depicts the bifurcation diagrams for the three different values of the damping coeIcient. If the length of
the upper body is encreased the limit cycles disappear because of scuIng regardless of the value of the damping coeIcient. 6. Type III gait One of the most important aspects of bipedal locomotion is that the biped should maintain an erect posture during locomotion. For the "ve link passive robot we were not able to detect the set of parameters which would lead to the erect posture during locomotion. The introduction of the dampers improved the stability of the robot, but the direction of the upper body was still downward. The introduction of the torsional springs at the hip joints allowed us to "nd the corresponding set of parameters for the stable gait with erect body posture. The Poincare map is presented in the Fig. 5. The Poincare map clearly shows the existence of the limit cycle. The "gure represents the bifurcation diagrams for di6erent values of spring sti6ness, choosing the upper body length as a bifurcation parameter. We observed the existence of two sets of equilibria in Fig. 5. The "rst set emerges for the higher range of sti6ness, 0:3 6 k 6 2:0 Nm=rad. The range was computed numerically up to one signi"cant digit. As can be observed from the "gure the upper body position for this range of equilibria is almost vertical at the heel strike. The second set of limit cycles takes place at a lower sti6ness range, 0:005 6 k 6 0:14 Nm=rad. The range was computed numerically up to three signi"cant digits. We also note that, for sti6er springs, the upper body lengths that lead to stable limit cycle get longer. For this set of limit cycles, we observe a period doubling bifurcation. We do not analyze the dynamics beyond this bifurcation point, because we are interested only in the period one cycle. (Our main purpose here is to identify limit cycles that will be useful in the control system analysis). Further increase of the upper body
626
E. Borzova, Y. Hurmuzlu / Automatica 40 (2004) 621 – 629 0.50
vertical angle (rad)
q1 (rad) (step n+1)
Max modulus of Floquet is 0.63 0.46
0.42
3.14 3.12 3.10 3.08 3.06 3.04
0.38 Basin of attraction without damping
c = 0.01
c = 0.05 c = 0.03 0.05 0.10
0.15 0.20 0.25 length of the upper body (m)
(b)
Bifurcation diagram for different values of "c"
0.34 0.34
0.38
(a)
0.42
0.46
0.50
q1 (rad) (step n) Poincare Map
vertical angle (rad)
Fig. 4. Type II gait: (a) Poincare map for l = 0:218 m; c = 0:01 Nm=(rad=sec) slope angle =0:0495 rad; (b) bifurcation diagram for di6erent values of “c”, slope angle =0:0495 rad.
1.0
bifurcation point
0.8 0.6 0.4 0.2 0.0 30
0.422
40
(b)
50 60 70 80 length of the upper body (m)
-3
90x10
Bifurcation Diagram k = 0.01
0.418
2.5 k = 0.0051 2.0
0.416
0.414 Max modulus of Floquet is 0.78 0.412 0.412
(a)
0.414
0.416
0.418
0.420
0.422
q 1 (rad) (step n)
vertical angle (rad)
q 1 (rad) (step n+1)
0.420
k = 0.03 1.5
k = 0.06 k = 0.09
1.0
k = 0.14
0.5 0.0 k = 0.3
Poincare Map
(c)
k = 0.5
k = 1.8 k = 0.9 k = 1.4 k = 2.0
0.05 0.10 0.15 0.20 0.25 0.30 length of the upper body (m)
Bifurcation diagram for different values of "k"
Fig. 5. Type III gait: (a) Poincare map for l = 0:062 m; k = 0:09 Nm=rad, slope angle =0:0495 rad; (b) bifurcation diagram for k = 0:09 Nm=rad, slope angle =0:0495 rad; (c) bifurcation diagram for di6erent values of “k”, slope angle =0:0495 rad.
length results in the quasi-periodic motion, which also lies outside the scope of this paper. 7. Type IV gait As shown before, the presence of the dampers improved stability characteristic of the gait, while the presence of the springs facilitates keeping the erect body posture during locomotion. Now, we combine springs and dampers to generate stable passive walking with the erect upper body posture. The Poincare map shows the existence of the limit cycle (see Fig. 6). However, the expansion in the basin of attraction was less signi"cant compared to Type II gait. The
rate of convergence to the "xed point has improved. Similar to the Type II gait the Poincare map was constructed for di6erent initial conditions. Fig. 6 presents the bifurcation diagrams for di6erent values of spring sti6ness and damping coeIcient. Similar to Type III gait we observed the existence of two sets of equilibria. For the higher range of sti6ness (0:2 6 k 6 2:0 Nm=rad) and lower damping coeIcient (0:0001 6 c 6 0:001 Nm=(rad=sec) the upper body position is also almost vertical at the heel strike similar to the corresponding case of the spring sti6ness in Type III gait. The introduction of linear dampers allowed us to decrease the vertical angle at the heel strike for k = 0:2 Nm=rad and c = 0:001 Nm=(rad=sec). We could not detect stable gait for k = 0:2 Nm=rad in Type III gait.
E. Borzova, Y. Hurmuzlu / Automatica 40 (2004) 621 – 629 0.50
2.0
0.45
Basin of attraction without damping
0.40
k = 0.04 c = 0.01
k = 0.09 c = 0.01
k = 0.1 c = 0.02
1.5
vertical angle (rad)
q1 (rad) (step n+1)
Max modulus of Floquet is 0.46
627
k = 0.2 c = 0.02
1.0
k = 1.4 c = 0.004
0.5 0.0
k = 0.3 c = 0.001
k = 0.3 c = 0.01
k = 1.8 c = 0.001
k = 0.5 c = 0.002 k = 2.0 c = 0.0001
-0.5
0.35
0.1
(b) 0.30 0.30
0.2
k = 0.2 c = 0.001
0.3
length of the upper body(m)
Bifurcation diagram for different values of "k and c" 0.35
(a)
0.40
0.45
0.50
q1 (rad) (step n)
Poincare Map
0.56 0.55 0.54 0.53 0.52 0.51 0.50
50
(a)
52
54
56
58
-3
vertical angle (rad)
progression velocity (m/s)
Fig. 6. Type IV gait: (a) Poincare map for k = 0:09 Nm=rad; c = 0:01 Nm=(rad=sec); l = 0:063 m, slope angle 0:0495 rad; (b) bifurcation diagram for di6erent values of “k” and “c”, slope angle 0:0495 rad.
50
60x10
slope angle (rad)
-0.04 -0.05 -0.06 -0.07 -0.08 -0.09 -0.10
(b)
52
54
56
58
-3
60x10
slope angle (rad)
step length (m)
0.39 0.38 0.37 0.36
1.41
(c)
1.42 1.43 frequency (1/s)
1.44
Fig. 7. Parameters of the passive gait for l = 0:27 m; k = 1:4 Nm=rad; c = 0:004 Nm=(rad=sec): (a) progression velocity v.s. slope angle; (b) vertical angle v.s. slope angle; (c) step length v.s. frequency.
The second set of equilibria corresponds to the following parameters: 0:04 6 k 6 0:3 Nm=rad; 0:01 6 c 6 0:02 Nm= (rad=sec). For this set of limit cycles we did not observe any period doubling bifurcations, as we observed in Type III gait. At the same time it was possible to increase the upper limit of the spring sti6ness range in the presence of the linear dampers from k =0:14 Nm=rad (see Type III gait) to the k = 0:3 Nm=rad, while decreasing the corresponding vertical angle at the heel strike. The lower bound of the spring sti6ness in this case decreased from k =0:005 Nm=rad (see Type III gait) to k = 0:04 Nm=rad. 8. E+ect of the walking surface inclination angle on Type IV gait Another informative parameter variation is the slope angle. The introduction of torsional springs allowed us to maintain the erect upper body posture, while the introduction of
linear dampers increased the convergence rate and the basin of attraction as it was shown above. Our objective here is to study the e6ect of varying the slope angle on the gait of the passive biped with springs and dampers. Introduction of the torsional springs and the linear dampers into the system allowed us to increase the slope by 23% while maintaining gait stability. The deviation of the upper body posture from the vertical line was clearly dependent on the slope angle with the angle of deviation increasing as the slope angle increases (Fig. 7). This walking pattern is similar to the way humans walk down a steep slope. A person typically tries to maintain his balance by deQecting his upper body posture in the direction opposite to that the locomotion. Fig. 7 depicts the variation of the progression velocity with respect to the slope angle. The progression velocity was calculated as the ratio of the step length and the duration of the step in seconds. The slope increase of 23% resulted in the progression velocity increase of 16%.
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E. Borzova, Y. Hurmuzlu / Automatica 40 (2004) 621 – 629
The relationship between the step length and the frequency of the step when the slope angle has increased can also provide interesting insights into the nature of the passive gait. The plot is presented in Fig. 7. As seen from the plot when the slope angle increases the biped takes more frequent and longer steps. The main factor in increasing the progression velocity is taking more frequent steps. This walking pattern is also similar to the way humans walk down a steep slope. A person typically tries to maintain his balance by taking more frequent steps.
9. Conclusions In this paper the dynamics of the "ve link model with knees and upper body was analyzed. We were successful in detecting three limit cycles that exhibit three distinct upper body motions. We investigated the structural stability of these cycles subject to variations in the upper body length. We determined that the variation of the upper body length caused the period doubling bifurcation in case of the spinning (counterclockwise) upper body. We demonstrated that the rate of convergence and the stability can be improved with addition of linear dampers in the hip joints of the model. Our investigation also showed that erect body posture is only achievable when torsional springs are placed in the hip joints. In this case the addition of the linear dampers also improved the convergence rate and stability. It was demonstrated that introduction of the torsional springs and the linear dampers into the system allowed us to increase the slope of the walking surface by 23% while maintaining the gait stability. In addition to that, the slope increase of 23% resulted in the progression velocity increase of 23%. The equations of motion were derived using Mathematica package. The numerical simulations were programmed in C++.
Appendix A. The dimensions, masses and moments of inertia are shown in Table 1.
Table 1 Dimensions, masses and moments of inertia Member
Mass Length Mom. of Iner. Center of mass x; y mi (kg) li (m) Ji (kg m2 ) (m)
Stance shank Stance thigh Upper body Swing thigh Swing shank
1.013 2.345 0.223 2.345 1.013
0.44 0.35 0.0589 0.35 0.44
0.039 0.023 0.004 0.023 0.039
x = 0:025; y = 0:1697 x = 0:0; y = 0:09 x = 0:0; y = 0:0589 x = 0:0; y = 0:09 x = 0:025; y = 0:1697
Appendix B. During locomotion two di6erent impacts were considered: knee strike and heel strike. Knee strike was calculated using the principle of generalized momenta. Heel strike leads not only to the discontinuities in the general velocities but there is a sudden exchange in the role of the swing and the stance side members (Hurmuzlu & Moskowitz, 1986; Hurmuzlu, 1993a, b). The following transformation was used to relate the generalized coordinates immediately before and after the impact: 1 1 1 1 1 0 0 0 0 1 − + q ; 0 0 0 1 0 q = (B.1) 0 0 1 0 0 0 1 0 0 0 where the subscript “−” and “+” denote quantities immediately before and after impact and switching, respectively. Impact equations for the heel strike were derived using the principles of linear and angular impulse and momentum. K(q+ ) 0 + R1 (q− ; q˙− ) q˙ −1 0 ; (B.2) ˜ = R2 (q− ; q˙− ) C(q+ ) Fr 0 −1 where q− ; q+ ; q˙− and q˙+ are pre- and post-contact joint positions and velocities, respectively. K is a 5 × 5; C is a 2 × 5; R1 is a 5 × 1 and R2 is a 2 × 1 matrix function.
T F˜ r = F˜ rx ; F˜ ry is the impulse at the contact point of the swing leg with the ground. K(q− )q˙+ represents the angular momentum of the system about the pivot after the impact, R1 (q− ; q˙− ) represents the angular momentum of the system about the same point before impact. C(q+ )q˙+ is the linear momentum of the system after impact and R2 (q+ ; q˙+ ) represents the linear momentum of the system before impact. Eqs. (B.1) and (B.2) were used to calculate the generalized coordinates and generalized velocities immediately after impact in terms of the quantities immediately before impact. For solution to be valid it is necessary to satisfy two additional conditions. The "rst condition is the no slip condition at the contact point. The second condition has to be satis"ed because of the assumption that the circular foot does not experience any impulsive forces immediately after impact: F˜ rx (B.3) ¡ ; F˜ ry y˙ +T ¿ 0;
(B.4)
where is the coeIcient of friction between the contact point and the ground surface and y˙ +T is the velocity of the contact point immediately after the impact.
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References Basmajian, J. (1976). The human bicycle. Biomechanics, 1, 297–302. Borzova, E., & Hurmuzlu, Y. (2002). Passively walking "ve link robot. Proceedings of the sixth biennial conference on engineering systems design and analysis, July 8–11, Istanbul, Turkey. Coleman, M., Garcia, M., Ruina, A., Camp, J., & Chatterjee, A. (1997). Stability and chaos in passive dynamic locomotion. Proceedings of the IUTAM symposium on new applications of nonlinear dynamics and chaos in mechanics, Cornell University, Ithaca, NY. Garcia, M. (1998). Stability, scaling and chaos in passive-dynamic gait models. Ph.D. thesis, Cornell University, Ithaca, NY. Garcia, M., Chatterjee, A., Ruina, A., & Coleman, M. (1998). The simplest walking model: Stability, complexity and scaling. ASME Journal of Biomechanical Engineering, 120(2), 281–288. Gomes, M., & Ruina, A. (2003). A non-singular way to add degrees of freedom: A homotopy method for "nding periodic motions of linked rigid body models. Proceedings of the nineth international symposium on computer simulation in biomechanics, July 2– 4, Sydney, Australia (to appear). Goswami, A., Thuilot, B., & Espiau, B. (1996). Compass-like biped robot, Part 1: Stability and bifurcation of passive gaits. INRIA Research Report, No.2996. Goswami, A., Thuilot, B., & Espiau, B. (1998). A study of the passive gait of a compass-like biped robot: Symmetry and chaos. The International Journal of Robotics Research, 17(12), 1282–1301. Hurmuzlu, Y. (1993a). Dynamics of bipedal gait: Part I—Objective functions and the contact event of a planar "ve-link biped. Journal of Applied Mechanics, 60, 331–336. Hurmuzlu, Y. (1993b). Dynamics of bipedal gait: Part II—Stability analysis of a planar "ve-link biped. Journal of Applied Mechanics, 60, 337–343. Hurmuzlu, Y., & Moskowitz, G. (1986). The role of impact in the stability of bipedal locomotion. Dynamics and Stability of Systems, 1, 217–234. Hurmuzlu, Y., & Moskowitz, G. (1987). Bipedal locomotion stabilized by impact and switching: I. Two and three dimensional, three element models, II. Structural stability analysis of a four element bipedal locomotion model. Dynamics and Stability of Systems, 2, 73–112. Kuo, A. (1999). Stabilization of lateral motion in passive dynamic walking. The International Journal of Robotics Research, 18(9), 917–930.
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Yildirim Hurmuzlu received his Ph.D. degree in Mechanical Engineering from Drexel University. Since 1987, he has been at the Southern Methodist University, Dallas, Texas, where he is a Professor and Chairman of the Department of Mechanical Engineering. His research focuses on nonlinear dynamical systems and Control, with emphasis on robotics, biomechanics, and vibration control. He has published more than 60 articles in these areas. Dr. Hurmuzlu is the associate Editor of the ASME Transactions on Dynamics Systems, Measurement and Control. Elena Borzova was born in Moscow, Russia. She received B.S., M.S. with honors from Moscow State University, Department of Mechanics and Mathematics in 1992. She received Ph.D. in Mechanical Engineering from Southern Methodist University in 2002. Currently she is working as an adjunct professor at Southern Methodist University. Dr. Borzova’s research interests include bipedal locomotion, nonlinear control, system dynamics and bifurcation theory.