Rigid vs compliant contact: an experimental study on biped walking

Abstract

Contact modeling plays a central role in motion planning, simulation and control of legged robots, as legged locomotion is realized through contact. The two prevailing approaches to model the contact consider rigid and compliant premise at interaction ports. Contrary to the dynamics model of legged systems with rigid contact (without impact) which is straightforward to develop, there is no consensus among researchers to employ a standard compliant contact model. Our main goal in this paper is to study the dynamics model structure of bipedal walking systems with rigid contact and a novel compliant contact model, and to present experimental validation of both models. For the model with rigid contact, after developing the model of the articulated bodies in flight phase without any contact with environment, we apply the holonomic constraints at contact points and develop a constrained dynamics model of the robot in both single and double support phases. For the model with compliant contact, we propose a novel nonlinear contact model and simulate motion of the robot using this model. In order to show the performance of the developed models, we compare obtained results from these models to the empirical measurements from bipedal walking of the human-size humanoid robot Surena III, which has been designed and fabricated at CAST, University of Tehran. This analysis shows the merit of both models in estimating dynamic behavior of the robot walking on a semi-rigid surface. The model with rigid contact, which is less complex and independent of the physical properties of the contacting bodies, can be employed for model-based motion optimization, analysis as well as control, while the model with compliant contact and more complexity is suitable for more realistic simulation scenarios.

Appendix: Drive system identification

1.1 A.1 Method

Our main goal in this section is to employ a simple representation which can replicate drive system dynamics behavior to a desired level of accuracy. The major effects that may be taken into account for identification of the drive system are the effective inertia, Coulomb friction, and viscous friction of the system and some other load-dependent terms [47]. Hence, the considered general model may be represented as

$$\begin{aligned} \tau _{\text{drive}}=j\ddot{\theta }+b \dot{\theta }+ c \dot{ \theta }^{3}+f \operatorname{sign}(\dot{\theta }) + \cdots , \end{aligned}$$

(32)

in which $j$ is the inertia parameter, $b$ and $c$ are the parameters for viscous friction (and potentially the electromotive force of the motor, back emf), and $f$ is the Coulomb friction parameter. Furthermore, $\tau _{\text{drive}}$ is the torque that is exerted by the motor, and $\theta$ is the joint angle which is measured by the encoder mounted at the output of the drive system.

In order to identify unknown parameters, a least squares curve fitting approach is adopted. The input trajectories that are considered for the identification procedure are the joints trajectories for various walking speeds. These trajectories include high and low velocity, as well as low and high frequency commands. Moreover, the motor torque can be computed by measuring the motor current and multiplying it to the motor torque constant, or exploiting a torque sensor at the output of the drive system. As a result, the linear regression model may be specified as

$$\begin{aligned} \tau _{m \times 1}= \begin{bmatrix} \ddot{\theta }& \dot{\theta }& \dot{\theta }^{3} & \operatorname{sign}( \dot{\theta }) \end{bmatrix} _{m \times n} \begin{bmatrix} j \\ b \\c \\ f \\ \vdots \end{bmatrix} _{n \times 1}, \end{aligned}$$

(33)

in which $m$ specifies samples that are taken from the measured values during one experiment, and $n$ is the number of parameters that should be identified. It should be noted that a necessary condition for identification is that $m$ should be greater than $n$. Using the Moore–Penrose inverse (left pseudo-inverse), the identification routine is carried out to minimize the quadratic norm of the parameters error

$$\begin{aligned} \begin{bmatrix} j \\ b \\c \\ f \\ \vdots \end{bmatrix} _{n \times 1}= \begin{bmatrix} \ddot{\theta }& \dot{\theta }& \dot{\theta }^{3} & \operatorname{sign}( \dot{\theta }) \end{bmatrix} ^{\dagger }_{n \times m} \; \tau _{m \times 1}. \end{aligned}$$

(34)

Using this method, for each experiment, a set of parameters may be obtained. As a result, in order to obtain a model which is valid for a wide range of experiments, the average value of obtained parameters may be considered as a candidate for the overall model. The obtained identified model will be acceptable, provided that it is valid for a wide range of experiments. This consistency can be evaluated using the consistency measure [47], which is the ratio of the standard deviation STDV to the average value AVG of each parameter, namely

$$\begin{aligned} \text{C.M.} = \frac{\text{STDV}}{\text{AVG}}. \end{aligned}$$

(35)

If the consistency measures obtained for all parameters are in a desired range [47], the obtained model is acceptable. Otherwise, some other terms should be added to the model to improve the consistency between the obtained parameters for various experiments.

1.2 A.2 Results

The drive system of the Surena III humanoid robot is composed of EC motors, pulleys and timing belts, and harmonic drive gears. In Fig. 13, the components of the drive system and developed test-stand for the identification purpose are shown. The three major effects that are taken into account in our identification routine are the effective inertia, Coulomb friction, and viscous friction of the system. Hence, the considered model may be represented as

$$\begin{aligned} \tau _{\text{drive}}=N_{p} N_{h} k_{m} i=j\ddot{\theta }+b \dot{\theta }+f \operatorname{sign}(\dot{\theta }), \end{aligned}$$

(36)

in which $N_{p}$ and $N_{h}$ are the pulley and harmonic reduction ratios, $k_{m}$ is the motor torque constant, and $i$ is the motor input current; $j$, $b$, and $f$ are the parameters that should be identified. It should be noted that since no torque sensor is available at the output of the harmonic drive, the load-dependent terms are not included in this model.

Fig. 13

The developed test-bed for system identification, system components (left), experimental setup (right)

Using the procedure that has been described in Sect. A.1, the identification routine is carried out and the obtained values for the identified model are summarized in Table 3. These values are obtained by applying 5 experiments using the knee joint motion for walking from 0.3 to 0.7 km/h.

Table 3 Obtained results from identification of the drive system, $j$ is the estimated inertia, $b$ is the estimated viscus friction coefficient, and $f$ is the estimated Coulomb friction coefficient

As it can be observed in Table 3, the obtained consistency measure for the parameter $f$ is absolutely acceptable. Also, for the parameters $j$ and $b$ this measure is satisfactory for our comparison purposes [47]. Therefore, the obtained model with average values moderately estimates the dynamics of the drive system for various walking speeds.

In Fig. 14, the identified model for the drive system of the knee joint is plotted. As it can be observed, the model fairly estimates the behavior of the drive system at the speed of 0.5 km/h. Also, in order to analyze effects of the components of the model, in Fig. 15 each effect is plotted separately. As it can be seen in this figure, the inertia and viscous friction have a dominant effect in high velocity and acceleration motions. However, the Coulomb friction effect has an approximately constant value which varies when the direction of motion changes.

Fig. 14

The obtained results for the knee joint identification

Fig. 15

Inertia, viscous and coulomb friction effects on the knee joint torque