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Abstract

This paper deals with the landing motion of an articulated legged robot. Humans use a peculiar crouching motion to land safely which can be characterized by body stiffness and damping. A stiffness controller formulation is used to realize this human behavior for the robot. Using this method, the landing motion is achieved with only the desired body stiffness and damping values, without desired COG(Center of Gravity) or joint paths. To achieve soft landing, variable body stiffness and damping values were optimized.

PBOT, which has four links with flexible joints was used for validation of the landing controller. A body stiffness and damping controller was used as an outer landing control loop and a fast subsystem controller for flexible joints was used as an inner force control loop. Simulations and experimental results about the landing motion are presented to show the performance of the body stiffness and damping controller.

Keywords

Legged locomotion Landing motion Hopping robot Body stiffness and damping Flexible joint

1. Introduction

In human motion analysis, a hopping motion is usually modeled as a simple spring-mass(Blickhan, R. (1989), Farley, C. T. & Gonzalez, O. (1996)). This model is realized in robotics through various experiments by compensating energy loss using bounce of leg by Raibert(Raibert, M. H. (1985)). After this, similar investigations were published(Gregorio, P.; Ahmadi, M. & Buehler, M. (1994), Berkermeier, M. D. & Desai, K. V. (1999)). Most of them used the identical mechanical structure with some advanced features in hopping method and in structure.

Running and hopping of the biped humanoid robot is now a very challenging area to make humanoid more agile and fast motion. There were some research efforts(Chevallereau, C.; Westervelt, E. R., & Grizzle, J. W. (2005), Park, J. H. & Kwon, O. (2003)) to make an articulated legged robot running by various path planning and nonlinear control techniques. In 2003, Sony realized running motion using biped humanoid robot QRIO(Nagasaka, K. et. al.; Kuroki, Y.; Suzuki, S.; Itoh, Y. & Yamaguchi, J. (2004)). In 2004, Honda demonstrated running in human size biped robot using ASIMO(Honda) and enhanced speed up to 6km/h. In (Nagasaka, K.; Kuroki, Y.; Suzuki, S.; Itoh, Y. & Yamaguchi, J. (2004)), method for acquiring analytical solution of the x-, y- COG path considering angular momentum using dynamic boundary conditions is presented for the running motion. In (Kajita, S.; Nagasaki, T.; Kaneko, K, Yokoi, K & Tanie, K. (2004)), the desired angular and linear momentum is calculated from a predetermined COG and the robot is controlled through the developed controller.

However, a limitation of these previous researches about humanoid was that controller follows desired COG path or dynamic boundary condition provided by the operator. This can produce unexpected foot touch-downs so that the COG path must be corrected to ensure motion stability because of the difficulty in predicting impact timing and robot state after impact. In the case of running and landing motion of humans, the characteristic named body stiffness and damping is used(Poul, E. B.; Poulsen, D. & Voigt, M. (1991)) to hop or land. It is known that a human behaves like a spring at the beginning of the impact and changes behavior into damping characteristics.

In this paper, a body stiffness and damping controller which resembles human's landing strategy will be applied to landing motion of legged robot, PBOT (Fig. 1) which has been made for hopping or running motion. The articulated legged robot system is assumed as a lumped single mass (COG) and forces from body stiffness and damping is applied on COG as a virtual force. Similar approach using body compliance was tried by (Ohashi, E. & Ohnishi, K. (2004)). But this method also follows desired path while requiring force and acceleration sensor signals to follow desired body impedance. The method developed in this paper utilizes simple structure of stiffness controller which needs only position and velocity signals with desired body stiffness and damping values.

Fig. 1.

Photo of PBOT

Target control motion is drop landing which starts motion from given height with standstill to the ground until robot stops motion. Constant and variable body stiffness and damping value case will be introduced and compared to each other. Variable value case shows soft landing behavior which resembles human behavior. Section. 2 describes PBOT specification. In section.3, we validate body stiffness and damping strategy using two mass system simulations. Section.4 introduces application processes of body stiffness and damping controller to the articulated legged model and fast subsystem controller. Section.5 shows experimental result of body stiffness and damping control with PBOT.

2. PBOT, Experimental Robot

PBOT (Fig. 1) is a four-link (foot, shank, thigh, torso) legged robot that has 3 actuators and 9 degrees of freedom (x; y; μ₀; μ₁; μ₂; μ₃; Z₁; Z₂; Z₃) for the planar motion, where Z_i is a flexible force coordinate (spring force) for ankle, knee and hip joint(belt force from belt deflection). Linear actuation structure is used for ankle and knee joint which uses series elastic actuator(Pratt, G. A. & Willianson, M. M. (1995)) where ball screw and spring (60kN/m) are serially connected with motor to allow shock tolerance about impact situation. Linear actuator is connected to the robot by forming closed kinematic chain with flexible joint around ankle and knee joint. Hip joint is actuated by a rotary motor with gear through belt transmission. The foot of PBOT is a planar foot which has four FSR (Force Sensing Resistor, used for touch sensing) at its corners. Beneath the foot there is rubber for shock absorption with 2.5mm thickness.

Encoders of 6 are installed to measure joint angles and linear displacement of the springs. Absolute angle of the foot is measured using INS (gyro) sensor. PBOT weighs 7.9kg and stands 0.95m tall. A dimension is based on small adult and detailed dimensions are in Table. 1.

Table 1.
Kinematic and mass data for PBOT

Link index(link i) 0 1 2 3 actuator1 actuator2

mass(kg, mi) 0.77 1.19 1.46 2.02 1.25 1.25

Inertia(kgm2, li) 0.01 0.02 0.03 0.01 0.01 0.01

Link Length(m, li) l0: 0.07 l01 :0.13 0.34 0.34 0.30 r1a1 :0.29 r1a2 : 0.29 r2a2 : 0.30

COG Length(m, ri) r01 : 0.13 r02 : 0.03 0.17 0.17 0.14 ra1 : variable ra2 : variable

Link index(link i)	0	1	2	3	actuator1	actuator2
mass(kg, mi)	0.77	1.19	1.46	2.02	1.25	1.25
Inertia(kgm2, li)	0.01	0.02	0.03	0.01	0.01	0.01
Link Length(m, li)	l0: 0.07	l01 :0.13	0.34	0.34	0.30	r1a1 :0.29	r1a2 : 0.29	r2a2 : 0.30
COG Length(m, ri)	r01 : 0.13	r02 : 0.03	0.17	0.17	0.14	ra1 : variable	ra2 : variable

3. Body Stiffness and Damping Controller

To make a landing strategy for a robot, it is important to understand human's landing strategy. Human's landing strategy has a relation with shock absorption for ground impact which can be divided into two categories. One is shock absorption by materials using muscles, tendons and foot heel pad. The other is shock absorption by motion like crouching which can be seen at the gymnastic competitions. Impact force is transformed into another energy form like potential energy (stored at the elastic material) or kinetic energy. In this paper, we use this human's motion behavior for shock absorption to make robot land on the ground.

To apply human's behavior to the robot, we need mathematical description of human's landing motion. From biomechanics study, a human's hopping and landing behavior was explained using parametric study. When the human hops continually, leg stiffness value increases from touch-down to take-off with the same initial value on every instants(Poul, E. B.; Poulsen, D. & Voigt, M. (1991)). When landing, the human also behaves like mass with spring-damper system. Initially, human acts like spring on landing but changes action into damping. Leg stiffness is expressed as

k_{leg} = \frac{F_{peak}}{c / L}

(1)

where F_peak is peak reaction force in the force platform when human jumps onto platform and ¢ L is a vertical displacement of body's center of mass.

Using this human's vertical spring and damper-like motion behavior, we apply similar strategy to legged robot landing control. Leg stiffness k_leg is changed by body stiffness k_leg which is expanded from leg motion to whole body motion and force platform force F_peak by F_g Which is a force on center of gravity transferred from the ground.

3.1. Application to Two-mass Hopper Model

In this section, one dimensional (vertical directional) model (Fig. 3) composed of two masses is used to see the effect of body stiffness and damping action on landing motion. Upper mass (7.15kg) represents lumped mass of the whole body (legged robot, humans, etc.) and lower mass (0.75kg) is used to describe impact phenomenon with the ground.

Fig. 2.

PBOT model description

Fig. 3.

1-D two mass system

To apply body stiffness, (X₁ - X₂) at impact time is used as a definition of equilibrium length for stiffness. Thrust force from body stiffness and damping is expressed as,

F_{t h r u s t} = K (x_{e q} i (x_{1} i x_{2})) i D (\underline{x_{2}} i \underline{x_{1}})

(2)

Fig.4(a) shows the result of drop landing simulation with constant values of K (2000N/m, stiffness of the system) and D (200Ns/m, damping of the system) at 0.1m initial height (X₂). We can see that the two-mass system has landed on the ground showing body stiffness and damping behavior on X₁. Landing was possible because spring element stored energy with upper mass support while damping element dissipated kinetic energy. Fig. 4(b) shows upper mass motions after landing which can is controlled through changing the stiffness and damping values.

Fig. 4.

1-D body stiffness and damping results

From human's case, it is known that body stiffness and damping behavior is changed from spring to damper which is not constant. If we see Fig. 5(b), we can see impulse thrust force from constant body stiffness and damping value case which is unrealistic thrust force in real experiment. Also thrust force is transmitting impulse force to the upper body. To make body stiffness and damping controller more feasible for landing motion, variable value case like human's behavior is needed.

Fig. 5.

Constant and variable K & D results comparison

In (Rapoport, S.; Mizarahi, J.; Kimmel, E.; Verbitsky, O. & Isakov, E. (2003)), the joint stiffness was found to be first-order in human hopping motion. And from Farley, C. T. & Morgenroth, D. C. (1999), we can assume body stiffness as first-order using the result of (Rapoport, S.; Mizarahi, J.; Kimmel, E.; Verbitsky, O. & Isakov, E. (2003))]. To define variable body stiffness and damping profiles, we assume the model of body stiffness and damping value as a first-order model as follows.

K = K_{0} + K_{1} t

(3)

D = D_{0} + D_{1} t

(4)

where t is a time after impact timing.

Fig. 5(a) shows comparison of upper mass position between constant and variable body stiffness and damping value cases. The motions of each case are similar in its fashion but required thrust force (Fig. 5(b)) is different from each other. Variable body stiffness and damping value case shows moderate increase of thrust force whereas constant value case shows impulsive force peak. At variable value case, zero initial body damping is used to prevent impulsive force transmission to upper body, while nonzero body stiffness is used to prevent excessive undershoot of upper mass at initial contact duration.

From the two mass drop landing simulation, it is identified that human's body stiffness and damping behavior can be used as a landing motion controller about the system described like Fig. 3.

4. Body Stiffness and Damping Control for Articulated Legged Model, PBOT

In this section, this humans's motion characteristic, i.e. body stiffness and damping behavior is applied to the landing strategy for the articulated legged robot system. First, definitions of body stiffness and damping on articulated legged model will be explained. And then, development processes from desired body stiffness and damping values to the controller formulation for flexible joint of PBOT are explained. Block diagrams in Fig. 6 helps understanding of the processes.

Fig. 6.

Control block diagram

4.1. Formulation of Body Stiffness and Damping Value for PBOT

Though there are actual springs in the actuator (at series elastic actuator), the stiffness of the spring is a fixed value. To assign arbitrary stiffness and damping value to the robot, a stiffness controller formulation is used here. The stiffness controller is a force controller to react against external force, but we use this controller to react against vertical COG motion (X₁ motion in section. 3) on landing motion. The task space in stiffness controller is changed into COG point. The stiffness controller equation is written as,

¿= J^{T} F; (F : external force)

(5)

where J is Jacobian about task coordinate of the robot and F is external force to the task point. To change into body stiffness and damping controller, the Jacobian is changed into COG Jacobian(Sugihara, T. & Nakamura, Y. (2002)) to decompose desired COG force(F_g) into joint space.

\begin{aligned} ¿ & = J_{g}^{T} F_{g} = J_{g}^{T} (F_{K} + F_{D}) \\ = J_{g}^{T} (K_{g} (X_{g e} ¡ X_{g}) ¡ D_{g} {\underline{X}}_{g}) \end{aligned}

(6)

X_g, (X_g;Y_g)^T is a COG position vector calculated from foot to the COG point. X_ge, (X_ge; Y_ge)^T is the equilibrium position of COG which is selected according to the desired motion after landing. Using this formulation, stiffness and damping forces used in section. 3 is replaced by virtual stiffness and damping forces transmitted to the COG. So, the articulated legged system can be regarded as a lumped mass with virtual spring and damper force.

4.1.1 Determining COG Jacobian

The COG Jacobian is calculated only when the foot is contact with the ground. If we determine J_g at global frame (coordinate attached at the ground at Fig. 2), J_g is changed according to contact mode (heel point contact, toe point contact and full line contact in 2-D case). On point contact case, contact point becomes under-actuated joint (passive joint) while nonzero desired torque is assigned by COG Jacobian (Eq. (6)) on that joint.

In this paper, to avoid from above mentioned case, we use a local frame which is attached at the foot (Fig. 2) as a reference coordinate for COG Jacobian. Desired COG force in global frame is projected into the local frame (Eq. (7)) to thrust robot into right direction. Flexible force coordinate (Z₁, Z₂, Z₃) is ignored in calculating COG Jacobian.

¿= J_{g}^{T} R (μ_{0}) F_{g}

(7)

where R(μ₀) is 2£ 2 rotation matrix around μ₀.

It is assumed that μ₀ is small enough and foot contact (point, line) is ensured through suppression role of the controller about contact point (it is known that suppression controller is good at decreasing pre-transition phase (Doh, N.; Chung, W. K.; & Youm, Y. (2000)), and the body stiffness and damping controller continuously suppresses contact point by thrusting force). In experiment, μ₀ is acquired through INS sensor which has 70Hz frequency enough for control. Drift problem of INS can be neglected because of short running time of landing experiment.

There are 3 actuator degrees of freedom (ankle, knee and hip) whereas the motion degrees of freedom is 2(X_g, y_g) on body stiffness and damping controller. To avoid redundant motion, torso pitch angle (μ₃) is added as a target control variable to make COG Jacobian 3 pound; 3 matrix.

4.1.2 Actuator Force Tracking Controller

Desired joint torques obtained from above section are transformed into desired actuator force on each actuator module using the derivation process of dynamics by closed chain kinematics characteristics(Ghorbel, F. H.; Chetelat, O.; Gunawardana, R. & Longchamp, R. (2000)) of model. To follow desired actuator forces in flexible joint system, force tracking controller is developed. All the joints of the PBOT are flexible joints which have transmission through springs and belt. To develop force tracking controller, we change dynamics in singularly perturbed form(Readman, M. C. (1994)) as,

where X_s is a slow variables, [x y μ₀ μ₁ μ₂ μ₃]^T and Z is a fast variable, [Z₁ Z₂ Z₃]^T which is a force in the spring(ankle, knee actuator) and belt(hip joint).¹ = 1 =k_s and k_s is a stiffness matrix for spring & belt. C₁, C₂ matrices comprise coriolis, centrifugal and gravity force. A is an input coeffcient matrix generated from closed chain kinematic characteristic of the system and B is a pre- multiplication matrix for input torque which is composed of gear and ball screw ratio.

Above dynamics equation can be divided into slow and fast subsystem. Slow subsystem (upper part of Eq. (8)) represents motion of the slow variables like rigid links and fast subsystem (lower part of Eq. (8)) represents motion of the spring and belt. Solving Eq. (8) for gives Solving Eq. (8) for gives

If we select control input U as

Eq. (11) changes into

To make the closed loop fast subsystem (Eq. (13)) asymptotically stable, all of the coeffcient matrix at the left-hand side must be and positive definite(Leang S. S.; Mohamad, M. M. & Hani, M. D. (1987)). Since ${M_{f}^{¡}}^{1}$ is positive definite because determinant of positive definite matrix(inertia matrix) is positive definite(Roger, A. H. & Charles, R. J. (1985)). If we select k_p and k_v so that the stability condition is satisfied, fast subsystem becomes asymptotically stable while following slow manifold, Z_d (C_f at the right hand side is composed of only slow variables). The force tracking controller can be considered as an inner force control loop for flexible joint structure of PBOT and the body stiffness and damping controller is regarded as an outer control loop (Fig. 6).

4.2 Drop Landing Simulation of PBOT

Developed body stiffness and damping controller will be used for drop landing simulation from given height. Simulation model is based on the parameters at Table. 2.

Table 2.
Landing simulation with 5˜10cm drop

5 cm drop x_g y_g Torso pitch

Stiffness 200N/m 2˜3000N/m 1.0N/m

Damping 100Ns/m 2˜300N/m 0.5Ns/m

10cm drop x_g y_g Torso pitch

Stiffness 200N/m 2000N/m 1.0N/m

Damping 100Ns/m 2˜300N/m 0.5Ns/m

4.2.1 Constant Body Stiffness and Damping Case

To validate developed body stiffness and damping controller, constant body stiffness and damping value case is first tried. y_ge is determined as a COG height at impact timing and x_ge as a foot center point. Fig. 7 shows y_g motion path with different two simulation environment (5cm, 10cm height) under body stiffness and damping values in Table. 2. Values for torso pitch are small because of high gear ratio at hip actuator. Fig. 7(a) is the case when foot is initially 5cm height parallel to the ground. y_g path follows body stiffness and damping behavior according to the assigned values. y_g value around impact time (0.15s) is relatively smooth without discontinuity which means soft landing. If we see force tracking result of ankle and knee actuator at Fig. 8 (K_g: 3000Nm, D_g : 200Ns/m), controller is following approximate desired force except right after the impact due to abrupt spring deflexion from impact force.

Fig. 7.

Yg for drop landing simulation

Fig. 8.

Desired & present actuator force and error for 5cm drop case

Large force tracking error at knee actuator right after the impact time is because ankle joint torque is highly coupled with knee joint torque, which can be identified from (A_f B)ⁱ ¹in Eq. (12). Ankle actuator force tracking affects knee actuator force tracking performance but not in vice versa. Hip joint control result is omitted because of little error due to high stiffness value at belt. Fig. 10 shows landing simulation scene from left to right direction.

Fig. 9.

Desired & present actuator force and error for 10cm drop case

Fig. 10.

Landing simulation of PBOT

Though we were not aware of the exact impact time, articulated legged robot was able to land with soft upper body motion by assigned body stiffness and damping values. If initial height is 10cm, y_g does not show smooth path (Fig. 7(b), Fig. 9) because of large impact force and actuator limit (0.2Nm). In case of initially slanted foot (μ₀=5^±), although there are separate impacts at the toe and heel, we can see that landing through body stiffness and damping is possible in Fig. 11.

Fig. 11.

Desired & present actuator force and y_g for 5cm drop with 5^± foot slope case

4.2.2 Variable Body Stiffness and Damping Case

Before applying variable body stiffness and damping value, it is important to know which values of coeffcients (Eq. 4) are suitable for landing motion. In case of constant value case, we can follow the processes in (Balandin, D. V.; Bolotnik, N. N. & Pilkey, W. D. (2001)). To determine the coefficients of body stiffness and damping value for variable case, multiobjective optimization with simple model (Fig. 3) is used to get cofficients according to given conditions. An Objective function for optimization can be decided as a thrust force peak minimization (because of actuator limit). Optimization constraints can be selected like these.

\begin{aligned} Minimize : & f : thrust force peak \\ Subject to : & g_{1} : x_{1} ¡ x_{2} > x_{lower} \\ g_{2} : F < F_{max} \\ g_{3} : {\underline{x}}_{1} > 0; t_{1} < t < t_{2} \\ g_{4} : x_{1} ¡ x_{2} < x_{st}; t_{1} < t < t_{2} \end{aligned}

where F_max is $M a x j F_{g} j = M a x j {(J_{g}^{T})}^{¡}^{1} ¿ j$ . g₁ is used to prevent undershoot of COG below joint limit. g₃ and g₄ are used to drive ${\underline{x}}_{1}$ to zero at steady state(x_st).

Initial positions for drop landing are x₁ = 4.7m, x₂ = 0.05m and time duration for stiffness and damping value change is confined as 0.5sec after initial impact which is a sufficient settlement time for landing.

The optimization result is in Fig. 12(a) for initial body stiffness value with 2000N/m. Both of the stiffness and damping values have been increased for landing motion. If initial stiffness value is large, stiffness value is decreased to diminish thrust force peak. On the contrary, small initial stiffness value causes increasing slope of body stiffness to prevent excessive undershoot (g₁ constraint). If we compare the result between simple and PBOT model, we can see similar but delayed y_g motion (5cm drop simulation) because of force tracking error (Fig. 12(b)).

Fig. 12.

Variable body stiffness and damping case

If we see force tracking result (Fig. 13), we can see that desired actuator force has been changed into moderate increase comparing with constant values case. In case of ankle, force tracking error decreased into small amount of value. But in knee joint, though desired actuator force has been changed into moderate increase, spring flexed abruptly because of impact force which is inevitable. Although optimized value does not guarantee the best result on simulation because of force control error, we can use this result as an initial guess for better landing performances.

Fig. 13.

variable body stiffness & damping force tracking

5. Experiment with PBOT

Experiment has been performed using PBOT with real time control loop (1kHz control frequency) using Pentium 650MHz computer. PBOT was dropped from given height to the ground (between 5» 10cm from the ground). All the data are collected right from the initial impact timing.

5.1 Constant Body Stiffness and Damping Case

Experiment with constant body stiffness and damping values has been done using four kinds of different values. In Fig. 14, we can see the experimental results of y_g for four cases of body stiffness and damping values. Fig. 14(a) is the case when K_g is 2000N/m and Fig. 14(b) for 3000N/m each with 200Ns/m and 300Ns/m body damping values.

Fig. 14.

Drop landing experiment for four cases

We can see that both of cases show the body stiffness and damping behavior on landing according to given body behavior. The difference of steady state y_g values between the simulation (Fig. 7) and experiment (Fig. 14) is because of different equilibrium length for body stiffness. Force tracking result when K_g is 2000N/m and D_g is 300Ns/m is at Fig. 15 which shows more force fluctuation due to hard contact environment at experiment (only 2.5mm rubber beneath the foot as a material shock absorption). On Fig. 18, we can see the sequential pictures of landing experiment from left to right direction.

Fig. 15.

Actuator forces, K_g: 2000N/m, D_g : 300Ns/m

5.2 Variable Body Stiffness and Damping Case

To see the effect of variable body stiffness and damping case, we used the result of Table. 3 which was calculated through the optimization process of section. 4. Initial body stiffness was selected as 1500N/m, 2000N/m and 2500N/m to see the various results with different values.

Table 3.
Body stiffness & damping values at experiments

K₀ K₁ D₁

Case I 1500N/m 1268 N/m 1388 Ns/m

Case II 2000N/m 1755 N/m 841 Ns/m

Case III 2500N/m 2636 N/m 967 Ns/m

	K₀	K₁	D₁
Case I	1500N/m	1268 N/m	1388 Ns/m
Case II	2000N/m	1755 N/m	841 Ns/m
Case III	2500N/m	2636 N/m	967 Ns/m

At Fig. 16, we can see the experimental and simulation results of y_g on various variable body stiffness and damping cases. Experimental results are similar to the theoretical behaviors of simulations according to the given values. Force tracking desired at Fig. 17 has moderate increase due to initial zero body damping force. Initial COG behavior right after the impact timing is smoother than at the constant cases which can be interpreted as more soft landing. Initial peaks of actuator force desired at ankle are due to the multiple impacts at the foot with the ground.

Fig. 16.

Drop landing for variable body stiffness & damping

Fig. 17.

Actuator force comparison of Case II in Table 3

Fig. 18.

Snapshots of landing experiment

6. Conclusion

This paper has presented body stiffness and damping controller for an articulated legged robot as a landing controller which resembles human's landing behavior. Body stiffness and damping behavior is realized through stiffness controller formulation about COG point and fast subsystem controller for flexible joints has been developed for PBOT model.

Through simulations and experiments, we have shown that landing motion for articulated legged system can be realized through body stiffness and damping control. We also have verified that variable body stiffness and damping case which is similar to human landing behavior is better for landing than constant value cases. Developed landing controller has an advantage that it does not require desired path and prior knowledge about impact time for landing motion but needs only desired body stiffness and damping values with position and velocity feedback.

Because we have used flat foot which has little damping materials beneath the foot, high frequency shock was not eliminated effectively which has produced fluctuation at actuator force control or rough force desired signal. To solve this problem, development of soft foot with arch structure is in work process. Also, developed body stiffness and damping controller is being applied to the development of controller for hopping motion of PBOT.

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