A Posture Balance Controller for a Humanoid Robot using State and Disturbance-Observer-Based State Feedback

Abstract

In order to use humanoid robots in our daily lives, stable robot walking is very important. This paper proposes a posture balance controller for a humanoid robot in order to achieve stable locomotion. The robot was modeled in simplified form as an inverted pendulum having a spring and a damper and the state feedback controller based on a disturbance and a state observer estimating the angle and angular velocity of the center of mass (COM) was developed with the simple model. Since a humanoid robot has different modeling parameters according to a number of the supporting legs and/or moving direction, four controllers were designed. With considering disturbance, the robot could estimated the state exactly and maintained the posture balance while disturbance is applied to the robot. The proposed controller was applied to a humanoid robot, DRC-HUBO2, and it was verified with some experiments in the lab and success of the stair mission in the DRC Finals 2015.

Appendix A: Controllers for experiment

1.1 A.1 Controller for DSP in the Sagittal Plane

The state space equation is as below:

$$\begin{array}{@{}rcl@{}} \left[\begin{array}{c} \dot{\theta} \\ \ddot{\theta} \\ \dot{F}_{d} \end{array}\right]&=&\left[\begin{array}{rrr} 0 & 1 & 0\\ -90.88 & -1.004 & 0.0173 \\ 0 & 0 & 0 \end{array}\right] \left[\begin{array}{c} {\theta} \\ \dot{\theta} \\ F_{d} \end{array}\right] \\ &+&\left[\begin{array}{r} 0 \\ 132.6 \\ 0\end{array}\right] u \end{array} $$

(30)

$$\begin{array}{@{}rcl@{}} y&=&\left[\begin{array}{ccc} 6.416 & 0.0623 & 0 \end{array}\right] \left[\begin{array}{c} {\theta} \\ \dot{\theta} \\ F_{d} \end{array}\right] \\ &+&\left[\begin{array}{c} -8.226 \end{array}\right] u. \end{array} $$

(31)

The observer gain and the controller gain are given below:

$$\begin{array}{@{}rcl@{}} L&=&\left[\begin{array}{r} 3.443 \\ 14.57 \\ 4606 \end{array}\right], \end{array} $$

(32)

$$\begin{array}{@{}rcl@{}} K&=&\left[\begin{array}{cc} -0.4966 & 0.0678 \end{array}\right]. \end{array} $$

(33)

The control law is as below:

$$\begin{array}{@{}rcl@{}} u&=&- \left[\begin{array}{cc} K & \frac{l^{2}}{k} \end{array}\right] \left[\begin{array}{r} {\theta} \\ \dot{\theta} \\ F_{d} \end{array}\right], \end{array} $$

(34)

where k is 4658 Nm/rad.

1.2 A.2 Controller for DSP in the Frontal Plane

The state space equation is as below:

$$\begin{array}{@{}rcl@{}} \left[\begin{array}{c} \dot{\theta} \\ \ddot{\theta} \\ \dot{F}_{d} \end{array}\right]&=&\left[\begin{array}{rrr} 0 & 1 & 0\\ -135.6 & -0.9594 & 0.0173 \\ 0 & 0 & 0 \end{array}\right] \left[\begin{array}{c} {\theta} \\ \dot{\theta} \\ F_{d} \end{array}\right] \\ &+&\left[\begin{array}{r} 0 \\ 190.0 \\ 0\end{array}\right] u \end{array} $$

(35)

$$\begin{array}{@{}rcl@{}} y&=&\left[\begin{array}{ccc} 9.191 & 0.0595 & 0 \end{array}\right] \left[\begin{array}{c} {\theta} \\ \dot{\theta} \\ F_{d} \end{array}\right] \\ &+&\left[\begin{array}{c} -11.78 \end{array}\right] u. \end{array} $$

(36)

The observer gain and the controller gain are given below:

$$\begin{array}{@{}rcl@{}} L&=&\left[\begin{array}{r} 2.471 \\ 5.574 \\ 3216 \end{array}\right], \end{array} $$

(37)

$$\begin{array}{@{}rcl@{}} K&=&\left[\begin{array}{cc} -0.5822 & 0.0476 \end{array}\right]. \end{array} $$

(38)

The control law is as below:

$$\begin{array}{@{}rcl@{}} u&=&- \left[\begin{array}{cc} K & \frac{l^{2}}{k} \end{array}\right] \left[\begin{array}{c} {\theta} \\ \dot{\theta} \\ F_{d} \end{array}\right], \end{array} $$

(39)

where k is 6672 Nm/rad.

1.3 A.3 Controller for SSP in the Sagittal Plane

The state space equation is as below:

$$\begin{array}{@{}rcl@{}} \left[\begin{array}{c} \dot{\theta} \\ \ddot{\theta} \\ \dot{F}_{d} \end{array}\right]&=&\left[\begin{array}{rrr} 0 & 1 & 0\\ -47.34 & -0.8610 & 0.0173 \\ 0 & 0 & 0 \end{array}\right] \left[\begin{array}{c} {\theta} \\ \dot{\theta} \\ F_{d} \end{array}\right] \\ &+&\left[\begin{array}{r} 0 \\ 76.82 \\ 0\end{array}\right] u \end{array} $$

(40)

$$\begin{array}{@{}rcl@{}} y&=&\left[\begin{array}{ccc} 3.716 & 0.05370 & 0 \end{array}\right] \left[\begin{array}{c} {\theta} \\ \dot{\theta} \\ F_{d} \end{array}\right] \\ &+&\left[\begin{array}{c} -4.764 \end{array}\right] u. \end{array} $$

(41)

The observer gain and the controller gain are given below:

$$\begin{array}{@{}rcl@{}} L&=&\left[\begin{array}{r} 5.707 \\ 35.90 \\ 7954 \end{array}\right], \end{array} $$

(42)

$$\begin{array}{@{}rcl@{}} K&=&\left[\begin{array}{cc} -0.2907 & 0.1189 \end{array}\right]. \end{array} $$

(43)

The control law is as below:

$$\begin{array}{@{}rcl@{}} u&=&- \left[\begin{array}{cc} K & \frac{l^{2}}{k} \end{array}\right] \left[\begin{array}{c} {\theta} \\ \dot{\theta} \\ F_{d} \end{array}\right], \end{array} $$

(44)

where k is 2698 Nm/rad.

1.4 A.4 Controller for SSP in the Frontal Plane

The state space equation is as below:

$$\begin{array}{@{}rcl@{}} \left[\begin{array}{c} \dot{\theta} \\ \ddot{\theta} \\ \dot{F}_{d} \end{array}\right]&=&\left[\begin{array}{rrr} 0 & 1 & 0\\ -25.06 & -0.8702 & 0.0173 \\ 0 & 0 & 0 \end{array}\right] \left[\begin{array}{c} {\theta} \\ \dot{\theta} \\ F_{d} \end{array}\right] \\ &+&\left[\begin{array}{r} 0 \\ 48.25 \\ 0\end{array}\right] u \end{array} $$

(45)

$$\begin{array}{@{}rcl@{}} y&=&\left[\begin{array}{ccc} 2.334 & 0.0540 & 0 \end{array}\right] \left[\begin{array}{c} {\theta} \\ \dot{\theta} \\ F_{d} \end{array}\right] \\ &+&\left[\begin{array}{c} -2.992 \end{array}\right] u. \end{array} $$

(46)

The observer gain and the controller gain are given below:

$$\begin{array}{@{}rcl@{}} L&=&\left[\begin{array}{r} 8.430 \\ 64.01 \\ 12660 \end{array}\right], \end{array} $$

(47)

$$\begin{array}{@{}rcl@{}} K&=&\left[\begin{array}{cc} -0.0010 & 0.1892 \end{array}\right]. \end{array} $$

(48)

The control law is as below:

$$\begin{array}{@{}rcl@{}} u&=&- \left[\begin{array}{cc} K & \frac{l^{2}}{k} \end{array}\right] \left[\begin{array}{c} {\theta} \\ \dot{\theta} \\ F_{d} \end{array}\right], \end{array} $$

(49)

where k is 1694 Nm/rad.