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This paper considers an obstacle avoidance control problem for the compass-type biped robot, especially circular obstacles are dealt with. First, a sufficient condition such that the swing leg does not collide the circular obstacle is derived. Next, an optimal control problem for the discrete compass-type robot is formulated and a solving method of the problem by the sequential quadratic programming is presented in order to calculate a discrete control input. Then, a transformation method that converts a discrete control input into a continuous zero-order hold input via discrete Lagrange-d’ Alembert principle is explained. From the results of numerical simulations, it turns out that obstacle avoidance control for the continuous compass-type robot can be achieved by the proposed method.

Humanoid robots have been energetically researched in the fields of robotics and control theory so far. Especially, the compass-type biped robot has been mainly studied as one of the simplest models of humanoid robots. For example, theoretical analysis of passive walking [

In this paper, we deal with an obstacle avoidance control problem for the compass-type biped robot via discrete mechanics. The contents of this paper is as follows. In Section 2, some fundamental concepts on discrete mechanics are summed up. Next, we derive the continuous and discrete compass-type biped robots based on both continuous and discrete mechanics, respectively in Section 3. Then, in Section 4, we formulate an obstacle avoidance control problem for the discrete compass-type biped robot and propose a solving method of it by the sequential quadratic programming to calculate a discrete control input. Furthermore, a transformation method from a discrete control input into a continuous zero-order hold input based on discrete Lagrange- d’Alembert principle is developed. In Section 5, we show some numerical simulations for the continuous compass-type biped robot in order to confirm the effectiveness of our method. Finally, we make a conclusion in Section 6.

This section summarizes fundamental concepts of discrete mechanics. See [

as the tangent space of Q at a point

where q is expressed as a internally dividing point of

and a discrete action sum:

Next, the discrete equations of motion is summarized. Consider a variation of points on Q as

The discrete Hamilton’s principle states that only a motion which makes the discrete action sum (3) stationary is realized. Calculating (4), we have

where

with the initial and terminal equations:

It turns out that (6) is represented as difference equations which contains three points

Then, we consider a method to add external forces to the discrete Euler-Lagrange equations. By an analogy of continuous mechanics, we denote discrete external forces by

where we define right/left discrete external forces:

respectively. By right/left discrete external forces, a continuous external force

Calculating variations for (8), we obtain the discrete Euler-Lagrange equations with discrete external forces:

with the initial and terminal equations:

In this subsection, we first give a problem setting of the compass-type biped robot. In this paper, we consider a simple compass-type biped robot which consists of two rigid bars (Leg 1 and 2) and a joint without rotational friction (Waist) as shown in

This subsection derives a model of continuous compass-type biped robot (CCBR) by using usual continuous mechanics. Denote the angles of Leg 1 and 2 by

We now derive a model of the CCBR. We assume that the torque at the waist can be controlled, and denote it by

Substituting the Lagrangian (13) into the Euler-Lagrange equations and adding the control input to the right-hand sides of them, we have the model of the CCBR as

Next, we derive a model of discrete compass-type biped robot (CCBR) via discrete mechanics in this subsection. We here use the notations; h: the sampling time;

In this paper, we use only the model of the DCBR in the swing phases, and hence we will derive it. By using the transformation law from a continuous Lagrangian into a discrete Lagrangian (2), we obtain the discrete Lagrangian as

from (13). Since the left and right discrete external forces (9) satisfy

Then, substituting (13) and (16) into the discrete Euler-Lagrange Equation (11) and the initial and terminal Equations (12), and adding the discrete control input (17) to these systems, we have the model of the DCBR as

It is noted that the detailed model of the CCBR can be derived by calculating (18)-(23) with (13) and (16).

In this subsection, we first give the problem setting of circular obstacles. As shown in

where

In addition, we set a point P as a desired grounding point for the swing leg of the robot. Based on the setting above, we consider the following problem on the gait generation for the compass-type biped robot.

Problem 1: For the continuous compass-type biped robot (CCBR) (14), (15), we find a control input v such that the swing leg of the CCBR lands at a desired grounding point P with avoiding collision with a circular obstacle

The initial and desired states of the CCBR are illustrated in

In order to solve Problem 1 above, a new method based on discrete mechanics will be developed. The method consists of the two steps: 1) calculation of a discrete control input by solving a finite dimensional constrained nonlinear optimization problem (Subsection 4.2); 2) transformation of a discrete control input into a zero-order hold input by discrete Lagrange-d’Alembert principle (Subsection 4.3).

Next, we consider an obstacle avoidance control problem of a discrete gait for the DCBR instead of the CCBR. The discrete obstacle avoidance control problem for the DCBR is stated as follows.

Problem 2: For the discrete compass-type biped robot (DCBR) (18)-(23), we find a sequence of the control input

Before formulation of Problem 2, we have to consider a condition on collision with a circular obstacle. The following theorem gives us a condition such that collision with a circular obstacle does not occur for the CCBR.

Theorem 1: If the next inequality

holds, then the swing leg of the CCBR does not collide the circular obstacle

(Proof) From

with a parameter s satisfying

When we implement the inequality condition (25) for numerical simulations, we have to make a modification for discretization. Hence, (25) is modified as

where S is the number of s. That is to say, we check the inequality condition for a finite number of s. Then, Problem 2 can be formulated as the following nonlinear optimization problem:

In the formulation above, (28) is a cost function on a sum of square of a discrete control input. We can see that the optimization control problem (28)-(30) is represented as a finite dimensional constrained nonlinear optimization problem with respect to the

In the previous subsection, we show a synthesis method of a discrete control input for the DCBR by solving a finite dimensional constrained nonlinear optimization problem. However, the obtained discrete control input cannot be utilized for the CCBR. So, we here consider transformation of a discrete control input into a con- tinuous one.

There exist infinite methods to generate a continuous control input from a given discrete one, and a conti- nuous control input generated from a given discrete input has to be consistent with laws of physics. Hence, in this paper, we deal with a zero-order hold input in the form:

which is one of the simplest continuous inputs. We need to derive a relationship between a discrete input

Theorem 2: A zero-order hold input (31) that satisfies discrete Lagrange-d’Alembert’s principle is given by

(Proof) For the time interval

we obtain

Hence, we can have (32).

By using (32) in Theorem 2, we can easily calculate a zero-order hold input from

In this section, we shall carry out some numerical simulations on continuous gait generation on slopes for the CCBR via the method proposed in the previous section, and confirm the effectiveness of our method. First, the problem setting is given. we set parameters as follows; parameters of the DCBR and the CCBR:

We set data of a circular obstacle as

Figures 6-9 show the simulation results.

This paper has developed a new approach to circular obstacle avoidance control for the compass-type biped robot from the view points of discrete mechanics and nonlinear optimization. Simulation results have shown that a gait that does not collide a circular obstacle can be generated, and hence we have confirmed the effectiveness of our new approach.

Our future work on control of humanoid robots via discrete mechanics includes the following themes: 1) extensions to various obstacles; 2) experimental evaluation of the proposed control method; 3) applications of discrete mechanics to more human-like robots and systems represented by partial differential equations.

TatsuyaKai, (2015) Circular Obstacle Avoidance Control of the Compass-Type Biped Robot Based on a Blending Method of Discrete Mechanics and Nonlinear Optimization. International Journal of Modern Nonlinear Theory and Application,04,179-189. doi: 10.4236/ijmnta.2015.43013