LAB

<latex>{\fontsize{16pt}\selectfont \textbf{Dynamically stable body path planning for quadruped locomotion}} </latex>

<latex>{\fontsize{12pt}\selectfont \textbf{[Fabian Tresoldi]}} </latex>
<latex>{\fontsize{10pt}\selectfont \textit{[Bachelor's thesis]}} </latex>

<latex> {\fontsize{12pt}\selectfont \textbf{Abstract} </latex>

The task of creating a motion plan for a 4-legged robot walking over irregular terrain can be split up into subproblems such as foothold selection, body path planning or swing leg trajectory planning. Given a set of current and future footholds and intermediate goals for the robots body between steps, the body path planning module is supposed to calculate a trajectory for the center of gravity (CoG) of the robots body which keeps the whole system stable. This project aims at implementing a planning method for the CoG trajectory based on quadratic programming (QP) and the dynamical zero-moment point (ZMP) stability criterion.

<latex> {\fontsize{12pt}\selectfont \textbf{ZMP stability criterion} </latex>

Statical stability of a quadruped is achieved by maintaining the projection of the CoG on flat ground within the support triangle formed by the stance feet. The ZMP criterion extends this by partly taking into account the acceleration of the robots body, allowing for higher speeds and developing the creeping gait associated with static stability into a more dynamical walking gait.
The ZMP marks the position of the point on the ground at which the reaction forces between the ground and robot do not produce a moment. With a simplified model of the robot one arrives at a closed-form expression for the ZMP depending linearly on the CoG position and acceleration. To keep the robot stable the ZMP has to be maintained within the support polygon formed by the stance feet, which is shrunken to account for assumptions in the model and execution errors.

<latex> {\fontsize{12pt}\selectfont \textbf{The QP problem} </latex>

To find a feasible CoG trajectory during a step as the solution of a QP problem the trajectory is described by multiple consecutive splines in $x$ and $y$ direction, thus representing the path of the CoG in a plane parallel to flat ground in terms of a vector of parameters $\boldsymbol{c}$. The requirements for the splined trajectory are chosen to be the following:

1. Minimize the integral of squared accelerations
2. Satisfy boundary conditions at the beginning and end of the step
3. Satisfy smoothness at spline switches
4. Satisfy the zero-moment point stability criterion

From requirement 1 follows the cost function $J(\boldsymbol{c})=\boldsymbol{c}^T A \boldsymbol{c}$ given by a quadratic form in $\boldsymbol{c}$. It is obtained by analytically differentiating the splines twice and summing up their integrated squares. Requirements 2 and 3 correspond to fixed values for the splines, which yield a set of linear equality constraints $D \boldsymbol{c} = \boldsymbol{d}$. Finally, the stability requirement is written as a set of linear inequalities $B \boldsymbol{c} < \boldsymbol{b}$. These are the result of evaluating the ZMP position at regular time intervals by using the ZMP model and the splined trajectory. Expressing the support polygon as three lines connecting the positions of the stance feet, the ZMP criterion leads to three inequalities per sample, repesenting each of the ZMP samples lying on either one or the other side of the lines.
The matrices and vectors $\{A, B, \boldsymbol{b}, D, \boldsymbol{d} \}$ constitute a constrained optimization problem known as quadratic programming, which can be solved by existing algorithms. Solving the QP problem provides the coefficients $\boldsymbol{c}$, which in turn define the desired CoG trajectory.

<latex> {\fontsize{12pt}\selectfont \textbf{Results} </latex>

The figures below show a trajectory calculated by the CoG path planning algorithm for a single step. The CoG is moved from standing still at the origin to a position $0.25m$ further ahead (in $y$-direction) in $1s$ with stance feet positions during the current and next step as shown in figure 2. The left front leg is active as the current swing leg, moving from $(x,y)=(-0.25m, 0.50m)$ to $(-0.25m, 0.75m)$ (with the right hind leg being next).

In figure 2 all instances of time up to the support polygon switch (putting down one swing leg and raising the next) are displayed in the left plot while all instances of time after the support polygon switch are displayed on the right. Remembering the ZMP criterion this allows to check the trajectory for feasibility. While the ZMP should remain within the shrunken support polygon formed by the current footholds in the left plot, it should at support polygon switch discretely jump to the shrunken support polygon formed by the next footholds in the right plot. This is achieved by deactivating the equality constraint enforcing smooth accelerations at that point in time.
The displacement, velocity and acceleration profiles for the same trajectory are shown in figure 1. It can be seen that the CoG reaches its goal and that the accelerations remain small. As intended, there is a discontinuity at $0.8s$ (the scheduled support polygon switch) for the ZMP to jump from one support polygon to the next. One easily recognizes that the ZMP criterion is violated by this trajectory.

<latex> {\fontsize{12pt}\selectfont \textbf{Conclusion} </latex>

The QP planning method for a dynamically stable CoG trajectory implemented in this work is modular and robot independent, making it easy to include it in a complete planning architecture for quadruped locomotion over rough terrain.
However, the faulty planning described above demonstrates that there are still unresolved issues with the implementation. After these problems have been solved, a next step could be to verify the method with a simple swing leg trajectory, regular foothold placement and simple orientation of the body. This would already allow the simulation of ZMP stable walking on flat ground.