Minimum-Time and Minimum-Jerk Gait Planning in Joint Space for Assistive Lower Limb Exoskeleton

Abstract

Assistive lower limb exoskeleton robot has been developed to help paraplegic patients walk again. A gait planning method of this robot must be able to plan a gait based on gait parameters, which can be changed during the stride according to human intention or walking conditions. The gait is usually planned in cartesian space, which has shortcomings such as singularities that may occur in inverse kinematics equations, and the angular velocity of the joints cannot be entered into the calculations. Therefore, it is vital to have a gait planning method in the joint space. In this paper, a minimum-time and minimum-jerk planner is proposed for the robot joints. To do so, a third-order system is defined, and the cost function is introduced to minimize the jerk of the joints throughout the stride. The minimum time required is calculated to keep the angular velocity trajectory within the range specified by the motor’s maximum speed. Boundary conditions of the joints are determined to secure backward balance and fulfill gait parameters. Finally, the proposed gait planning method is tested by its implementation on the Exoped^® exoskeleton.

Appendix: Inverted pendulum model

Considering the origin of the coordinates $\left(x,y\right)$ at the pivot point of the inverted pendulum shown in Fig. 5, the position and the linear velocity of CoM can be written as

$$\begin{array}{*{20}c} {x_{CoM} = - l\sin \left( {\theta_{h - sup} } \right){ ,}} & {\dot{x}_{CoM} = - l\dot{\theta }_{h - sup} \cos \left( {\theta_{h - sup} } \right)} \\ {y_{CoM} = l\cos \left( {\theta_{h - sup} } \right){ ,}} & {\dot{y}_{CoM} = - l\dot{\theta }_{h - sup} \sin \left( {\theta_{h - sup} } \right),} \\ \end{array}$$

(A.1)

where ${\dot{\theta }}_{h-sup}$ is the angular velocity. The dynamic model of the pendulum can be obtained easily as follows:

$$\ddot{\theta }_{h - sup} - w_{0}^{2} \theta_{h - sup} = 0{ ; }\sin \left( {\theta_{h - sup} } \right) \approx \theta_{h - sup} ,$$

(A.2)

where $w_{0} = \sqrt{\frac{g}{l}}$ is the natural frequency of the pendulum, and $\mathrm{g}$ is the acceleration of gravity. Solving Eq. (A.2) with an initial angle ${\theta }_{h-sup}\left({t}_{0}\right)$ and angular velocity ${\dot{\theta }}_{h-sup}\left({t}_{0}\right)$ gives Eq. (A.3)

$$\theta_{h - sup} \left( t \right) = \theta_{h - sup} \left( {t_{0} } \right)\left( {\frac{{e^{{w_{0} \left( {t - t_{0} } \right)}} + e^{{ - w_{0} \left( {t - t_{0} } \right)}} }}{2}} \right) + \frac{{\dot{\theta }_{h - sup} \left( {t_{0} } \right)}}{{w_{0} }}\left( {\frac{{e^{{w_{0} \left( {t - t_{0} } \right)}} - e^{{ - w_{0} \left( {t - t_{0} } \right)}} }}{2}} \right).$$

(A.3)

The initial angular velocity of the inverted pendulum ${\dot{\theta }}_{h-sup}\left({t}_{0}\right)$ that guides the pendulum from the initial angle ${\theta }_{h-sup}\left({t}_{0}\right)$ to the final angle ${\theta }_{h-sup}\left({t}_{f}\right)=0$ (CoM at the vertical line) with zero angular velocity during a period $\left({t}_{f}-{t}_{0}\right)$ can be obtained as Eq. (A.4) by substituting ${\theta }_{h-sup}\left(t\right)=0$ at $t={t}_{f}$ in Eq. (A.3)

$$\dot{\theta }_{h - sup} \left( {t_{0} } \right) = - \theta_{h - sup} \left( {t_{0} } \right)\left( {\frac{{e^{{w_{0} \left( {t_{f} - t_{0} } \right)}} + e^{{ - w_{0} \left( {t_{f} - t_{0} } \right)}} }}{{e^{{w_{0} \left( {t_{f} - t_{0} } \right)}} - e^{{ - w_{0} \left( {t_{f} - t_{0} } \right)}} }}} \right)w_{0} .$$

(A.4)