Dynamic Modeling and Control Study of the NAO Biped Robot with Improved Trajectory Planning

Abstract

Motion study of bipedal robots necessitates correct solutions of the forward and inverse kinematics with optimized and fast closed form computations which justifies an accurate kinematic model. On the other hand, dynamic modeling and stability analysis are essential for control study of humanoid robots to reach robust walk. This chapter is focused on dynamic modeling of the Nao humanoid robot, made by Aldebaran Co., in the RoboCup standard platform league. Moreover, trajectory approximation with a cubic Spline and kinematic analysis are described in brief here in this chapter. Main constraints such as inertial forces and joint angles for the given position and nominal conditions are simulated, mathematically described, and verified through experimental results from the real robot sensory data. The above mentioned modifications on the solution together with the dedication of other physical properties in dynamic modeling results in more precise acceleration and torque values as it is concluded in this work.

Appendix

Position constraints of foot and torso in sagittal plane are defined as:

$$ x_{f} \left( t \right) = \left\{ \begin{gathered} \left( {k - 1} \right)L_{step} \hfill \\ \left( {k - 1} \right)L_{step} \hfill \\ \left( {k - 1} \right)L_{step} + L_{\max } \hfill \\ (k + 1)L_{step} \hfill \\ \end{gathered} \right.\begin{array}{*{20}c} {,t = t_{1} } \\ {,t = t_{2} } \\ {,t = t_{3} } \\ {,t = t_{4} } \\ \end{array} \quad z_{f} \left( t \right) = \left\{ {\begin{array}{*{20}c} {FootHeight} & {t = t_{1} } \\ {FootHeight} & {t = t_{2} } \\ {h_{f\max } } & {t = t_{3} } \\ {FootHeight} & {t = t_{4} } \\ \end{array} } \right. $$

(A.1)

In which $L_{step}$ is the step length, $L_{\max }$ is the maximum horizontal distance of the ankle from the start point in $T_{\max } ,\,h_{f\max }$ is the maximum ankle height during $T_{step} .$ Constraints in torso position for specified times are as below:

$$ x_{t} \left( t \right) = \left\{ \begin{gathered} kL_{step} - 1.3x_{ts} \hfill \\ kL_{step} - x_{ts} \hfill \\ kL_{step} + x_{te} \hfill \\ \end{gathered} \right.\begin{array}{*{20}c} {,t = t_{1} } \\ {,t = t_{2} } \\ {,t = t_{4} } \\ \end{array} ,\,y_{t} = \left\{ {\begin{array}{*{20}c} {y_{tmid} } & {,t = t_{1} } \\ {y_{t\min } } & {,t = t_{3} } \\ \end{array} } \right.,\,z_{t} \left( t \right) = \left\{ {\begin{array}{*{20}c} {h_{t\min } } & {,t = t_{1} } \\ {h_{t\max } } & {,t = t_{3} } \\ {h_{t\min } } & {,t = t_{4} } \\ \end{array} } \right. $$

(A.2)

$y_{tmid}$ stands for the distance between the feet and $y_{t\min }$ is the minimum distance from the ankle of the supporting foot to the spinal column. Experimental results substantiate the margin of $y_{t\min }$ between $- 0.2\,y_{tmid}$ and $0.4\,y_{tmid} .$ Furthermore, $h_{t\max }$ and $h_{t\min }$ symbolizes maximum and minimum torso height (Tables 3, 4.

Table 3 Link vectors and CoM vectors extracted from the Nao physical specifications provided by Aldebaran Co. and geometrical modeling by MRL-Nao team

Table 4 The mass and diagonal elements of the inertia tensors of each Nao’s link

Approximation functions of Eq. 9 in text are described as:

$$ f_{1} (\theta_{3} ) = a_{3} (c\theta_{3} ) + d_{4} s\alpha_{3} (s\theta_{3} ) + a_{2} $$

(A.3)

$$ f_{2} (\theta_{3} ) = a_{3} c\alpha_{2} (s\theta_{3} ) - d_{4} s\alpha_{3} c\alpha_{2} (c\theta_{3} ) - d_{4} s\alpha_{2} c\alpha_{3} - d_{3} s\alpha_{2} $$

(A.4)

$$ f_{3} (\theta_{3} ) = a_{3} s\alpha_{2} (s\theta_{3} ) - d_{4} s\alpha_{3} s\alpha_{2} (c\theta_{3} ) + d_{4} c\alpha_{2} c\alpha_{3} + d_{3} c\alpha_{2} $$

(A.5)

$$ g_{1} (\theta_{2} ,\theta_{3} ) = \left[ {c\theta_{2} f_{1} (\theta_{3} )} \right] - \left[ {s\theta_{2} f_{2} (\theta_{3} )} \right] + a_{1} $$

(A.6)

$$ g_{2} (\theta_{2} ,\theta_{3} ) = c\alpha_{1} \left[ {s\theta_{2} f_{1} (\theta_{3} )} \right] + c\alpha_{1} \left[ {c\theta_{2} f_{2} (\theta_{3} )} \right] - s\alpha_{1} \left[ {f_{3} (\theta_{3} )} \right] - d_{2} s\alpha_{1} $$

(A.7)

$$ g_{3} (\theta_{2} ,\theta_{3} ) = s\alpha_{1} \left[ {s\theta_{2} f_{1} (\theta_{3} )} \right] + s\alpha_{1} \left[ {c\theta_{2} f_{2} (\theta_{3} )} \right] + c\alpha_{1} \left[ {f_{3} (\theta_{3} )} \right] + d_{2} c\alpha_{1} $$

(A.8)