Latest revision as of 11:15, 8 September 2023

Abstract

The energy optimal motion planning of a hopping robot with three links is investigated in the flight phase. Firstly, the conservation equation of angular momentum of the hopping robot in the flight phase is established which is a nonholonomic constraint. Then the energy consumption of the robot in the flight phase is selected as the optimization goal. Given the initial and terminal positions, the Gaussian pseudospectrum method is used to solve the optimal control problem. The simulation results show that the initial angular momentum has a great influence on the performance of the hopping robot. With the zero initial angular momentum, although the flight time can be selected arbitrarily, the greater the flight time, the smaller the energy consumption, the force required by the robot is greater. Thus, it is necessary to select an appropriate value.

Keywords: Hopping robot, flight phase, nonholonomic constraint, optimal motion planning

1. Introduction

Compared with wheeled or tracked robots, hopping robots move in a jumping manner and are capable of crossing obstacles which are several times their own size. They use discrete landing points to make contact with the ground and have strong adaptability to complex and unstructured terrain. The study of hopping robots has become a research focus in recent years [1]. Among them, Professor Raibert’s team has made groundbreaking research in the field of hopping robots, and their designed model of a single legged telescopic hopping robot provides a reference for subsequent research [2-6].

Before controlling the robot, it is necessary to plan the motion trajectory of the hopping robot [7-11]. Vermeulen [7] planned the robot based on its target motion parameters and used a quintic polynomial to describe the robot’s trajectory. Wu et al. [9]studied hopping robots with flat feet, using Bezier curves to represent the trajectory of active joints, and planning the trajectory based on minimizing driving energy.

In the flight phase, the robot foot is released from the ground. At this time, the only external force on the robot is gravity. The robot meets the principle of conservation of angular momentum, and is a nonholonomic constraint system. Due to the non integrability of nonholonomic constraints, its motion planning problem is much more complex than that of general systems. Rehman et al. [12] used a time-varying feedback control strategy to plan the motion during the flight phase. Guo et al. [13] used the direct single shot method to plan the jumping gait of a robot with four links, and its flight phase was almost passive.

The most commonly used motion planning methods for nonholonomic system can be divided into two categories, namely, the direct method and the indirect method [14-16]. The indirect method is based on the maximum (minimum) value principle, which transforms the optimal motion planning problem into a two-point boundary value problem. Its advantage is that the local optimal solution can be found, but it is difficult to guess the initial solution, and the radius of convergence is small. The direct method uses the parametric method to transform the optimal motion planning problem of the continuous system into a nonholonomic motion planning problem, and then obtains the optimal motion trajectory by solving the non motion planning problem. This method does not need to solve the first order optimal condition, and the radius of convergence is large. Pseudospectral method is the most widely used method among them.

In this paper, the optimal motion planning of a hopping robot with three links in the flight phase is studied. First, the constraint equation of the hopping robot in the flight phase is established according to the conservation principle of angular momentum. Then, the energy optimal motion planning problem is transformed into a nonholonomic motion planning problem by using the Gaussian pseudospectral method, and the problem is solved. The simulation results demonstrate the effectiveness of this method. Because the initial angular momentum has a great impact on the hopping robot. This paper finally analyzes the impact of the initial angular momentum on the hopping performance of the robot.

2. Dynamic model

The kinematics model of the robot in the flight phase is shown in Figure 1. The hopping robot consists of three components: body represented by 1, legs denoted by 2, and feet, namely, 3. In order to simplify the analysis, it is assumed that the motion of the hopping robot is limited to the sagittal plane, without considering the lateral motion. Assuming the length of each member of the robot is ${\displaystyle l_{i}\left(i=1,2,3\right)}$, the mass is mi, the position of the centroid Si of the body is ${\displaystyle AS_{i}}$, and ${\displaystyle l_{Si}={\alpha }_{i}l_{i}}$ (${\displaystyle {\alpha }_{i}}$ is the proportion factor of each centroid position), and ${\displaystyle 0<{\alpha }_{i}<{1}}$. The rotational inertia of each member around the center of mass is ${\displaystyle I_{i}}$.

Based on a floating coordinate with hip joint position A as the origin, the dynamic model is established. The generalized coordinate of the robot is ${\displaystyle x={\left[{\theta }_{0},{\theta }_{12},{\theta }_{23}\right]}^{T}}$, where ${\displaystyle {\theta }_{1}}$ is the absolute coordinate of the body, ${\displaystyle {\theta }_{12}}$ is the relative angle between the body and legs (counterclockwise is positive), namely, ${\displaystyle {\theta }_{12}={\theta }_{1}-{\theta }_{2}}$, ${\displaystyle {\theta }_{23}}$ is the relative angle between the legs and feet, i.e. ${\displaystyle {\theta }_{23}={\theta }_{2}-{\theta }_{\mbox{3}}}$.

Figure 1. Model of robot during flight phase

During the flight phase, the influence of air drag and internal friction of the robot on the robot’s hopping motion is ignored, and the robot’s total center of mass moves in a parabolic motion with the gravity. If the flight time is ${\textstyle T}$, then the horizontal and vertical motion of the center of mass can be written as

where, ${\displaystyle x_{c}{}^{to}}$ and ${\displaystyle y_{c}{}^{to}}$ are the horizontal and vertical positions of the robot’s total center of mass at the flight time, respectively. ${\displaystyle {\dot {x}}_{c}^{to}}$ and ${\displaystyle {\dot {y}}_{c}^{to}}$ are the horizontal and vertical velocities of the total center of mass at the flight time, respectively. The position of the robot at the touchdown is completely determined by the flight conditions.

The rotation of the robot can be expressed by the angular momentum of the robot with regarding to the total center of mass. Angular momentum can be calculated as follows:

where, ${\displaystyle {\overline {GG}}_{i}={\boldsymbol {r}}_{i}-{\boldsymbol {r}}_{G}}$, ${\displaystyle i={1},...,{3}}$, ${\displaystyle r_{i}}$ is the position of the centroid of each component, and ${\displaystyle {\boldsymbol {r}}_{G}}$ is the position of the total centroid.

The only external force on the robot in the flight phase is gravity. So the robot meets the principle of conservation of angular momentum about the total center of mass. Supposing that the angular momentum of the robot at the initial moment of the flight phase is ${\textstyle P_{0}}$, one can obtain:

where ${\displaystyle {\Delta }_{i}}$ is a function of joint rotation angle. The equation above indicates that the system is a nonholonomic system with Pfaffian constraints, and it has a cyclic coordinate ${\displaystyle {\theta }_{1}}$.

During the flight phase, the robot is only equipped with actuators in the hip and ankle joints. So the joint angular velocity ${\displaystyle {\dot {\theta }}_{12}}$ and ${\displaystyle {\dot {\theta }}_{23}}$ are selected as the control inputs for the system. Eq.(3) can be written in state space form. When ${\displaystyle P_{0}}$ do not equal 0, it is an affine nonlinear system with three states and two inputs and drift terms

where， ${\displaystyle {\boldsymbol {u}}={\left[{\begin{array}{cc}u_{1}&u_{2}\end{array}}\right]}^{T}={\left[{\begin{array}{cc}{\dot {\theta }}_{12}&{\dot {\theta }}_{23}\end{array}}\right]}^{T}}$.

When the initial angular momentum of the system is not zero, ${\displaystyle f\left(x\right)\not =0}$, then the system has no balance point.

The optimal trajectory planning of a robot refers to the initial position ${\displaystyle x_{0}={\left[{\theta }_{0},{\theta }_{\mbox{12}},{\theta }_{\mbox{23}}\right]}^{\mbox{T}}}$ and terminal position ${\displaystyle x_{T}={\left[q_{0d},q_{1d},q_{2d}\right]}^{T}}$ of the given robot, by searching for a set of control inputs ${\displaystyle u\left(t\right)}$ to enable the robot to move along a path that satisfies incomplete constraints. Meanwhile, the robot can move from the initial position ${\displaystyle x_{0}}$ to the terminal position ${\displaystyle x_{T}}$ within a given time ${\textstyle T}$, and achieve an extreme performance index.

In robot systems, due to the limited energy that robots can carry, in order to make the robot move for a longer time, it is necessary to consider the energy consumption of the robot. According to the minimum energy consumption theorem, in this paper the energy consumed by the hopping robot during flight phase is selected as the optimal control objective. Due to the fact that the usual driving method for robots is motor, and the energy consumed by the motor is proportional to its velocity, the performance index function can be expressed as follows:

In the flight phase, the optimal planning problem of a hopping robot can be described as: finding suitable control input variables to achieve the minimum performance index, i.e. Eq.(5). Meanwhile, the state variable and initial time ${\displaystyle t_{0}}$ and flight time ${\textstyle T}$ satisfy the following constraints:

where ${\displaystyle t_{0}}$ and T are constant.

3. Pseudospectral method for solving optimal control problems

In recent years, the pseudospectral method has become one of the important methods for solving optimal control problems [12]. The basic principle of the pseudospectral method is to discretize the continuous optimal control problem at the orthogonal collocation points, and approximate the state and control variables through the global interpolation polynomial. Thus the optimal control problem can be transformed into a nonlinear programming problem (NLP).

According to the different orthogonal collocation points, there are four widely used pseudospectral methods: Gaussian pseudospectral method, Radau pseudospectral method, Legendre pseudospectral method and Chebyshev pseudospectral method. Legendre pseudospectral method has the advantages of convergence speed of exponential function, insensitivity to initial value guess, and large radius of convergence. So Legendre pseudospectral method is adopted in this paper.

The Legendre pseudospectral method uses the roots of orthogonal polynomials as collocation points and global orthogonal polynomials as finite bases. The system’s state and control variables are discretized at the Lendre Guass Lobatto (LGL) point. The Lagrange interpolation polynomial is used to approximate the state and control variables, obtaining the discrete dynamic equations at the points. Thus, the differential operation in the state equation and the integral operation in the performance index function are transformed into algebraic operation, and finally the optimal control problem is transformed into a nonlinear programming problem with the state variables and control variables at the points as the parameters to be optimized.

Dynamic equation:

Boundary condition:

Inequality path constraint

where ${\displaystyle J\in R}$ is the performance indicator, ${\displaystyle \varphi \in R}$ is the Mayer type performance indicator, ${\displaystyle G\in R}$ is the Lagrange type performance indicator, and ${\displaystyle f\left(\cdot \right)}$ is the state equation function vector, ${\displaystyle \phi }$ is the initial and terminal constraint function vector, and ${\displaystyle C}$ is the equation and inequality path constraint function vector.

The calculation process of Legendre pseudospectral method is as follows: first, the value range of the original planning problem is mapped from ${\displaystyle t\in \left[t_{0},T_{f}\right]}$ to the distribution interval of discrete points in the pseudospectral method through time domain transformation ${\displaystyle \tau \in \left[-{1},{1}\right]}$.

The Legendre orthogonal polynomial on the interval ${\displaystyle \left[-{\mbox{1}},{\mbox{1}}\right]}$ is

where ${\displaystyle L_{n}\left(\tau \right)}$ represents n-order Legendre polynomials.

In order to achieve better interpolation approximation, ${\displaystyle {\tau }_{0}=-{1}}$, ${\displaystyle {\tau }_{N}={1}}$ and ${\displaystyle n-1}$ zeros ${\displaystyle {\tau }_{i}}$ ${\displaystyle (i={1},{2},\ldots ,{N-1})}$ of the first derivative of the Legendre polynomial ${\displaystyle {\dot {L}}_{n}\left(\tau \right)}$ is defined as an LGL point, there are ${\displaystyle N+1}$ control points, denoted as ${\displaystyle {\tau }_{i}}$ ${\displaystyle \left({i}={1},{2},\ldots ,{N}\right)}$.

Using ${\displaystyle n+1}$ LGL points as interpolation points, n-order Lagrangian interpolation polynomials is constructed to approximate the continuous state variable ${\displaystyle x\left(t\right)}$ and control variable ${\displaystyle u\left(t\right)}$, and obtain the following equations:

where, ${\displaystyle {\Phi }_{i}\left(\tau \right)}$ is the ${\displaystyle n}$-order Lagrangian interpolation basis function,

where, ${\displaystyle {i}={1},{2},\ldots \ldots ,n}$, ${\displaystyle {\Phi }_{i}\left(\tau \right)}$ satisfies the relationship ${\displaystyle {\Phi }_{i}\left(\tau \right)={\delta }_{ij}}$, if ${\displaystyle i=j}$, there is ${\displaystyle {\delta }_{ij}={1}}$； if ${\displaystyle i\not =j}$, ${\displaystyle {\delta }_{ij}={0}}$.

After parameterizing the state variable through interpolation polynomials, the differential operation of the state equation can be approximated as the differential operation of the interpolation basis function, and the derivative of the state variable ${\displaystyle x\left(\tau \right)}$ can be approximated as follows:

where ${\displaystyle k=0,{1},\cdots \cdots ,N}$, ${\displaystyle D}$ is the ${\displaystyle (n+1)}$ differential matrix, representing the differential value of Lagrange basis function at each LGL control point, ${\displaystyle D_{ki}}$ is the ${\displaystyle (k,i)}$th element. One can obtain

Now, the constraints of the state equation can be transformed into discrete state equations at N+1 LGL control points.

The boundary and path constraints after discretization can be described as follows:

where ${\displaystyle k=0,{1},\cdots ,N}$.

Through Gauss-Lobatto numerical integration, the integral term in the performance index function is transformed into the following algebraic expression:

where, ${\displaystyle w_{i}\left(i=0,{1},\cdots ,N\right)}$ is Gauss weights, defining

Through the above discretization and approximation processing, the optimal control problem can be further described by Eqs.(7)-(9) as follows: solving the values of the state variable ${\displaystyle X_{i}}$ at ${\displaystyle N+1}$ interpolation points ${\displaystyle X=(X_{0},X_{1},\cdots ,X_{N})}$ and the control variable ${\displaystyle U_{i}}$ at ${\displaystyle N+1}$ interpolation points ${\displaystyle U=(U_{0},U_{1},\cdots ,U_{N})}$, so that the performance index function takes the minimum value and satisfies the constraint conditions of Eqs. (16)-(18).