Minimum-Acceleration Trajectory Optimization for Humanoid Manipulator Based on Differential Evolution

Abstract

A humanoid manipulator produces significantly reactive forces against a humanoid body when it operates in a rapid and continuous reaction environment (e.g., playing baseball, ping-pong etc.). This not only disturbs the balance and stability of the humanoid robot, but also influences its operation precision. To solve this problem, a novel approach, which is able to generate a minimum-acceleration and continuous acceleration trajectory for the humanoid manipulator, is presented in this paper. By this method, the whole trajectory of humanoid manipulation is divided into two processes, i.e., the operation process and the return process. Moreover, the target operation point is considered as a particular point that should be passed through. As such, the trajectory of each process is described through a quartic polynomial in the joint space, after which the trajectory planning problem for the humanoid manipulator can be formulated as a global constrained optimization problem. In order to alleviate the reactive force, a fitness function that aims to minimize the maximum acceleration of each joint of the manipulator is defined, while differential evolution is employed to determine the joint accelerations of the target operation point. Thus, a trajectory with a minimum-acceleration and continuous acceleration profile is obtained, which can reduce the effect on the body and be favourable for the balance and stability of the humanoid robot to a certain extent. Finally, a humanoid robot with a 7-DOF manipulator for ping-pong playing is employed as an example. Simulation experiment results show the effectiveness of this method for the trajectory planning problem being studied.

Keywords

Trajectory Optimization Minimum-acceleration Trajectory Humanoid Manipulator Differential Evolution

1. Introduction

The trajectory planning of a humanoid manipulator is a fundamental research issue in humanoid robotics and is very important for robots serving the human[1, 2]. As for the trajectory planning problem of the manipulator, there are two different methods to solve this: one is Cartesian space planning and the other is joint space planning. Since each of the path points in the Cartesian space ought to be mapped into a set of joint angles, with these sets of joint angles interpolated with smooth functions within all the kinematic constraints of the manipulator[3], Cartesian space planning should involve a large amount of inverse kinematics computation[4].

As for a humanoid manipulator in a rapid and continuous reaction environment for high-speed objects, the goal assignment is that, in a given time, the end effector of the manipulator is moved from the static initial point to the target operation point with a required Cartesian speed to operate the target object, and then returns to the static initial point preparation for the next target operation[5, 6]. Due to the rod shape and the mechanism design, each joint of the humanoid manipulator has its own position constraint. Moreover, the manipulator's mechatronics system is limited by the size of its actuators, so the manipulator also has its joint velocity and acceleration limitation[3]. Hence, the trajectory of the manipulator should be planned within all the kinematic constraints (joint position, velocity, acceleration etc.); otherwise the planned trajectory cannot be achieved[7, 8]. Over the last couple of decades, many researchers have made efforts to address the relevant constraints problems. Wang et al. presented a smooth, minimum-acceleration trajectory planning (MATP) method within the velocity and acceleration constraints of a humanoid robot, which can find MATP from the arbitrary initial state to the arbitrary target state within a determined time frame[3]. Chan and Dubey used a weighted least-norm solution to avoid joint limits for redundant joint manipulators in order to guarantee joint limit avoidance and minimize unnecessary self-motion comparison with the gradient projection method[9]. Xiang et al. presented a general weighted least-norm (GWLN) method for the control of redundant manipulators, in which an experiment on the 7-DOF redundant manipulator illustrated that it can both guarantee the obstacle-free zone and not violate the joint limits in contrast to the directional gradient projection method[10]. The approach taken by Piazzi and Visioli, who developed a new approach based on interval analysis in order to find the global minimum-jerk trajectory of a robot manipulator within a joint space scheme, successfully exploited a variety of real automation settings[11].

Despite the aforementioned research, little has been done to deal with the operation problems of a humanoid robot in a rapid and continuous reaction environment. Ren et al. have proposed a trajectory planning method for a 7-DOF humanoid manipulator in a rapid and continuous reaction and obstacle avoidance environment, in which the employed method served to solve the trajectory planning problem of the humanoid manipulator for ping-pong playing [5]. However, this work did not consider the impact on the humanoid body as a consequence of the humanoid manipulator being subject to a rapid operation. Due to the dynamic coupling between the humanoid manipulator and the humanoid body (or unfixed base), the humanoid manipulator should produce a great reactive force against the body, especially when the manipulator operates rapidly in relation to high-speed objects. The reactive force against the humanoid body not only disturbs the balance and stability of the humanoid robot, but also significantly influences the operation precision of the humanoid manipulator. Hence, discovering how to alleviate the reactive force against the humanoid body ought to be considered when planning the motion trajectory of the humanoid manipulator.

With regard to the reactive force against the humanoid body, the acceleration of the manipulator's movement trajectory should be a critical factor that essentially disturbs the humanoid body, while a minimum-acceleration trajectory is favourable for keeping the balance and reducing its impact on the humanoid body[3, 12]. The continuity of acceleration is another important factor that influences the disturbance of the manipulator to the humanoid body. A discontinuous acceleration trajectory might cause the manipulator system to produce a hidden vibration during the rapid movement process of the manipulator, which not only seriously harms the humanoid manipulator, but also disturbs the humanoid robot greatly[12]. Generally, trajectory optimization is achieved by minimizing a suitable performance index on the basis of satisfy the assignment requirement within all the kinematic constraints. As such, the acceleration of the trajectory has been considered as a performance index for a humanoid manipulator operating in a rapid and continuous reaction environment. In this paper, a trajectory optimization method for a humanoid manipulator, based on differential evolution (DE), is presented in order to alleviate the reactive force against the humanoid body. It divides the whole trajectory of the humanoid manipulation into two stages, i.e., the operation process and the return process, with the target operation point being a particular point that should be passed through. Moreover, a quartic polynomial is adopted in order to interpolate these two processes in the joint space to maintain the continuous acceleration trajectory. In addition, a fitness function that aims to minimize the maximum acceleration of the trajectory is constituted, while the joint acceleration values at the target operation moment are determined through DE. By obtaining a trajectory with a minimum-acceleration and continuous acceleration profile, this approach method is more efficient for the trajectory planning problem.

2. Problem Description

For a robot system, the movement trajectory of this robot can be denoted by the change relationship $θ (t)$ between the joint position vector $θ$ and the time t. Then, $\dot{θ} (t)$ and $\overset{\cdot\cdot}{θ} (t)$ respectively represent the joint velocity trajectory and the joint acceleration trajectory of the robot system. The trajectory planning problem of a humanoid manipulator in a rapid and continuous reaction environment for high-speed objects is concerned with finding a movement trajectory that runs from the static initial state $[θ_{0}, 0]$ to the target operation state $[θ_{f}, {\dot{θ}}_{f}]$ within a determined time frame $t_{f 1}$ , all of which is defined as the operation process stage; and then returns to the static initial point from the target operation point within another determined time frame $t_{f 2}$ , all of which is defined as the return process stage. Moreover, the whole trajectory has a continuous acceleration profile and satisfies all the kinematic constraints. Simultaneously, the optimal minimum-acceleration trajectory should be found from all candidate trajectories that satisfy the requirements above; namely, the obtained trajectory, which is the one whose maximum acceleration of the trajectory is the minimum. The problem facing a humanoid manipulator for high-speed objects can be described as follows:

With reference to an n -DOF manipulator of the humanoid robot, suppose that the joint position of the static initial point is denoted by $θ_{0} = [θ_{10}, θ_{20}, \dots, θ_{n 0}]$ , the joint position of the target operation point at $t_{f 1}$ moment is represented as $θ_{f} = [θ_{1 f}, θ_{2 f}, \dots, θ_{n f}]$ , and the velocity of the target operation point at $t_{f 1}$ moment is denoted by ${\dot{θ}}_{f} = [{\dot{θ}}_{1 f}, {\dot{θ}}_{2 f}, \dots, {\dot{θ}}_{n f}]$ . The trajectory planning problem of the humanoid manipulator for high-speed objects can then be transformed into a trajectory optimization problem from all candidate trajectories of the system, which could be expressed by the following mathematical description:

θ^{*} (t) = arg min F (θ)

(1)

subject to

\begin{array}{l} θ^{*} (0) = θ_{0} ​ θ^{*} (t_{f 1}) = θ_{f} \\ {\overset{\cdot}{θ}}^{*} (0) = 0 ​ ​ ​ ​ ​ ​ ​ {\overset{\cdot}{θ}}^{*} (t_{f 1}) = {\overset{\cdot}{θ}}_{f} \end{array} (Operation process)

(2)

\begin{array}{l} θ^{*} (t_{f 1}) = θ_{f} θ^{*} (t_{f 1} + t_{f 2}) = θ_{0} \\ {\overset{\cdot}{θ}}^{*} (t_{f 1}) = {\overset{\cdot}{θ}}_{f} {\overset{\cdot}{θ}}^{*} (t_{f 1} + t_{f 2}) = 0 \end{array} (Return process)

(3)

lim_{t \to t_{0}} {\overset{\cdot\cdot}{θ}}_{i} (t) = {\overset{\cdot\cdot}{θ}}_{i} (t_{0}) ​ ​ \forall t_{0} \in [0, t_{f 1} + t_{f 2}], ​ ​ i = 1,2, \dots, n

(4)

where $F (θ)$ is an objective function that can minimize the maximum acceleration of the trajectory, $θ^{*} (t)$ is the selected optimum trajectory, $t_{f 1}$ is the execution time at the stage of operation process, $t_{f 2}$ is the determined time at the return process stage; meanwhile, formula (2) and (3) respectively correspond to the boundary conditions of the operation process and the return process, whereas formula (4) indicates the demand of the trajectory with continuous acceleration properties at any moment. In addition, the selected optimum trajectory should meet the following kinematic constraint conditions of the humanoid manipulator:

\begin{array}{l} θ_{i}^{min} \leq θ_{i}^{*} (t) \leq θ_{i}^{max} \forall t \in [0, t_{f 1} + t_{f 2}] \\ {\overset{\cdot}{θ}}_{i}^{min} \leq {\overset{\cdot}{θ}}_{i}^{*} (t) \leq {\overset{\cdot}{θ}}_{i}^{max} \forall t \in [0, t_{f 1} + t_{f 2}] \\ {\overset{\cdot\cdot}{θ}}_{i}^{min} \leq {\overset{\cdot\cdot}{θ}}_{i}^{*} (t) \leq {\overset{\cdot\cdot}{θ}}_{i}^{max} \forall t \in [0, t_{f 1} + t_{f 2}] \end{array}

(5)

where $i = 1,2,…, n$ , $θ_{i}^{min}$ and $θ_{i}^{max}$ are respectively the lower bound and the upper bound of the joint i position, ${\dot{θ}}_{i}^{min}$ and ${\dot{θ}}_{i}^{max}$ are respectively the lower bound and the upper bound of the joint i velocity, and ${\overset{\cdot\cdot}{θ}}_{i}^{min}$ and ${\overset{\cdot\cdot}{θ}}_{i}^{max}$ are respectively the lower bound and the upper bound of the joint i acceleration.

3. Motion Planning Strategy

As shown in Figure 1, when the manipulator operating in a rapid and continuous reaction environment, the whole trajectory is connected by two segments. The target point is given as a particular point that should be passed through between the operation process and the return process. Given that each segment is a point-to-point trajectory planning problem, the trajectory of the manipulator can be planned in the joint space.

Figure 1.

The operation process and the return process of the trajectory of the humanoid manipulator, where the solid line denotes the operation process and the dotted line represents the return process

According to the boundary conditions in formulae (2) and (3), Ren et al. adopted a cubic polynomial to interpolate these two processes[5]. However, this planning strategy ought to produce a discontinuous point of the acceleration trajectory at $t_{f 1}$ moment, which would probably lead to hidden vibrations in the humanoid manipulator and disturb the humanoid body greatly. In order to avoid this phenomenon, a quadrinomial polynomial planning strategy is used to interpolate the joint trajectory of these two processes.

At the stage of the operation process, the manipulator moves from the static initial point to the target operation point at $t_{f 1}$ moment. Hence, the boundary conditions of the joint position meet $θ (0) = θ_{0}$ , $θ (t_{f 1}) = θ_{f}$ , while the boundary conditions of the joint velocity satisfy $\dot{θ} (0) = 0$ , $\dot{θ} (t_{f 1}) = {\dot{θ}}_{f}$ . Assuming that the joint acceleration at $t_{f 1}$ moment is $\overset{\cdot\cdot}{θ} (t_{f 1}) = {\overset{\cdot\cdot}{θ}}_{f}$ , the joint trajectory of the operation process can be described by a quartic polynomial as follows:

\begin{array}{l} θ_{i} (t) = a_{i 0} + a_{i 1} t + a_{i 2} t^{2} + a_{i 3} t^{3} + a_{i 4} t^{4} \\ t \in [0, t_{f 1}] ​ ​ ​ i = 1,2, \dots, n \end{array}

(6)

where $a_{i 0}, a_{i 1}, \dots, a_{i 4}$ are constants of the polynomial. The boundary constraint conditions of the operation process are given as:

\begin{array}{l} θ_{i} (0) = a_{i 0} \\ θ_{i} (t_{f 1}) = a_{i 0} + a_{i 1} t_{f 1} + a_{i 2} t_{f 1}^{2} + a_{i 3} t_{f 1}^{3} + a_{i 4} t_{f 1}^{4} \\ {\overset{\cdot}{θ}}_{i} (0) = a_{i 1} \\ {\overset{\cdot}{θ}}_{i} (t_{f 1}) = a_{i 1} + 2 a_{i 2} t_{f 1} + 3 a_{i 3} t_{f 1}^{2} + 4 a_{i 4} t_{f 1}^{3} \\ {\overset{\cdot\cdot}{θ}}_{i} (t_{f 1}) = 2 a_{i 2} + 6 a_{i 3} t_{f 1} + 12 a_{i 4} t_{f 1}^{2} \end{array}

(7)

Hence, the five unknown constants can be solved as:

\begin{array}{l} a_{i 0} = θ_{i 0} ​ a_{i 1} = 0 \\ a_{i 2} = \frac{6}{t_{f 1}^{2}} (θ_{i f} - θ_{i 0}) - \frac{3}{t_{f 1}} {\overset{\cdot}{θ}}_{i f} + \frac{{\overset{\cdot\cdot}{θ}}_{i f}}{2} \\ a_{i 3} = - \frac{8}{t_{f 1}^{3}} (θ_{i f} - θ_{i 0}) + \frac{5}{t_{f 1}^{2}} {\overset{\cdot}{θ}}_{i f} - \frac{{\overset{\cdot\cdot}{θ}}_{i f}}{t_{f 1}} \\ a_{i 4} = \frac{3}{t_{f 1}^{4}} (θ_{i f} - θ_{i 0}) - \frac{2}{t_{f 1}^{3}} {\overset{\cdot}{θ}}_{i f} + \frac{{\overset{\cdot\cdot}{θ}}_{i f}}{2 t_{f 1}^{2}} \end{array}

(8)

At the stage of the return process, the manipulator returns to the static initial position from the target operation position at the $t_{f 1} + t_{f 2}$ moment, while the velocity of each joint returns to zero at this moment. Therefore, at this stage, the boundary conditions of the joint position satisfy $θ (t_{f 1}) = θ_{f}$ , $θ (t_{f 1} + t_{f 2}) = θ_{0}$ , while the boundary conditions of the joint velocity meet $\dot{θ} (t_{f 1}) = {\dot{θ}}_{f}$ , $\dot{θ} (t_{f 1} + t_{f 2}) = 0$ . If the joint acceleration at the $t_{f 1}$ moment is $\overset{\cdot\cdot}{θ} (t_{f 1}) = {\overset{\cdot\cdot}{θ}}_{f}$ , then the trajectory of the return process can be depicted by a quartic polynomial as follows[13]:

\begin{array}{l} θ_{i} (t) = b_{i 0} + b_{i 1} (t - t_{f 1}) + b_{i 2} {(t - t_{f 1})}^{2} + b_{i 3} {(t - t_{f 1})}^{3} \\ + b_{i 4} {(t - t_{f 1})}^{4} \\ t \in [t_{f 1}, t_{f 1} + t_{f 2}], ​ ​ ​ i = 1,2, \dots, n \end{array}

(9)

where $b_{i 0}, b_{i 1}, \dots, b_{i 4}$ are constants of the polynomial. The boundary constraint conditions of the return process are given as:

\begin{array}{l} θ_{i} (t_{f 1}) = b_{i 0} \\ θ_{i} (t_{f 1} + t_{f 2}) = b_{i 0} + b_{i 1} t_{f 2} + b_{i 2} t_{f 2}^{2} + b_{i 3} t_{f 2}^{3} + b_{i 4} t_{f 2}^{4} \\ {\overset{\cdot}{θ}}_{i} (t_{f 1}) = b_{i 1} \\ {\overset{\cdot}{θ}}_{i} (t_{f 1} + t_{f 2}) = b_{i 1} + 2 b_{i 2} t_{f 2} + 3 b_{i 3} t_{f 2}^{2} + 4 b_{i 4} t_{f 2}^{3} \\ {\overset{\cdot\cdot}{θ}}_{i} (t_{f 1}) = 2 b_{i 2} \end{array}

(10)

From these, we can determine that the five unknown constants are:

\begin{array}{l} b_{i 0} = θ_{i f} b_{i 1} = {\overset{\cdot}{θ}}_{i f} b_{i 2} = \frac{{\overset{\cdot\cdot}{θ}}_{i f}}{2} \\ b_{i 3} = (4 θ_{i 0} - 4 θ_{i f} - 3 {\overset{\cdot}{θ}}_{i f} t_{f 2} - {\overset{\cdot\cdot}{θ}}_{i f} t_{f 2}^{2}) / t_{f 2}^{3} \\ b_{i 4} = (- 3 θ_{i 0} + 3 θ_{i f} + 2 {\overset{\cdot}{θ}}_{i f} t_{f 2} + {\overset{\cdot\cdot}{θ}}_{i f} \frac{t_{f 2}^{2}}{2}) / t_{f 2}^{4} \end{array}

(11)

As formulated above, the total parameters that need to be determined concern the joint acceleration of the target operation position at the $t_{f 1}$ moment. For an n -DOF humanoid manipulator, there are n parameters to be defined. All of these parameters can be located by using the following optimization method.

4. Trajectory Optimization Based on Differential Evolution

4.1. Differential evolution

DE is a population-based intelligent search approach that solves the optimization problem through individuals' cooperation and competition[14]. In each iteration, DE implements differential mutation and crossover operators on the current population in order to produce a temporary population, and then employs a greedy selection procedure among the two populations in order to choose the best one-to-one.

For an n -dimensional optimization problem, suppose the population size is m, in which case the DE/best/1/bin mutation operator is performed on the current individual $x_{i}^{t}$ according to the following equation in order to firstly produce the mutant vector $v_{i}^{t}$ :

v_{i}^{t} = x_{g b e s t}^{t} + F (x_{r 1}^{t} - x_{r 2}^{t})

(12)

where $r 1, r 2 \in {1,2, \dots, m}$ are the randomly chosen indices at the tth iteration and $r 1 \neq r 2 \neq i$ , $x_{g b e s t}^{t}$ is the best individual of the current population, and $F \in [0,2]$ represents the scaling factor that is used to control the amount of perturbation in the process. Based on the mutant vector, a trial vector $u_{i}^{t}$ is constructed through a crossover operation, which combines components from the population vector $x_{i}^{t}$ and its corresponding mutant vector $v_{i}^{t}$ as follows:

u_{i j}^{t} = {\begin{array}{l} v_{i j}^{t} ​ r a n d (\cdot) \leq C R | j = r a n d n \\ x_{i j}^{t} ​ o t h e r w i s e \end{array}

(13)

where $r a n d (\cdot)$ is a uniform number in range [0,1], $C R$ is the crossover probability and $r a n d n$ is a randomly chosen integer within the set ${1,2,…, m}$ . Finally, the fitness of the vector $x_{i}^{t}$ and $u_{i}^{t}$ is compared, after which the better is chosen to generate offspring through greedy selection, as follows:

x_{i}^{t + 1} = {\begin{array}{l} u_{i}^{t} f (u_{i}^{t}) ​ s u p e r i o r t o f (x_{i}^{t}) \\ x_{i}^{t} o t h e r w i s e \end{array}

(14)

where $f (.)$ is the fitness function of DE. More details on the DE algorithm can be found in [14].

4.2. Optimization scheme of the trajectory

The rapid movement of the humanoid manipulator disturbs the balance and stability of the humanoid robot; meanwhile, the imbalance of the humanoid body influences the operation precision of the manipulator end-effector. Since the trajectory acceleration is an important factor for reducing its disturbance to the humanoid body in humanoid manipulator systems, an optimum trajectory with minimum-acceleration and continuous acceleration properties should be selected.

As mentioned previously, n parameters should be optimized for an n-DOF humanoid manipulator, that is $[{\overset{\cdot\cdot}{θ}}_{1 f}, {\overset{\cdot\cdot}{θ}}_{2 f}, \dots, {\overset{\cdot\cdot}{θ}}_{i f}, \dots, {\overset{\cdot\cdot}{θ}}_{n f}]$ , where ${\overset{\cdot\cdot}{θ}}_{i f} (i = 1,2, \dots, n)$ is the joint i acceleration of the target operation point at the $t_{f 1}$ moment. As DE[14] has a good computational efficiency and a simple concept, and can also be implemented easily, it has garnered much interest and led to a wide range of applications within robotics[15, 16]. A flowchart of the DE algorithm for selecting an optimum trajectory of an n-DOF humanoid manipulator is presented in Figure 2.

Figure 2.

Trajectory optimization of an n -DOF humanoid manipulator based on DE

In order to select an optimum trajectory of the humanoid manipulator for high-speed objects using DE, the fitness function of the DE algorithm ought to be determined. According to the optimization goal and request of formula (1), the fitness function of the DE algorithm can be defined as follows:

\begin{array}{l} min f (K) = \sum_{i = 1}^{n} w_{i} \cdot max | {\overset{\cdot\cdot}{θ}}_{i} (t) | \\ s . t . θ_{i} (t) \in [θ_{i}^{min}, θ_{i}^{max}] \\ {\overset{\cdot}{θ}}_{i} (t) \in [{\overset{\cdot}{θ}}_{i}^{min}, {\overset{\cdot}{θ}}_{i}^{max}] \\ {\overset{\cdot\cdot}{θ}}_{i} (t) \in [{\overset{\cdot\cdot}{θ}}_{i}^{min}, {\overset{\cdot\cdot}{θ}}_{i}^{max}] \\ t \in [0, t_{f 1} + t_{f 2}] ​ ​ ​ ​ i = 1,2, \dots, n \end{array}

(15)

where $K$ represents the individual in relation to DE and w_i represents the weight parameters of the joint i. Generally, compared with the wrist joint, the shoulder and elbow joints have a greater influence on the balance and stability of the humanoid body when the humanoid manipulator is operating rapidly. Hence, the weight parameters w_i of the shoulder and elbow joints should be set relatively larger. In addition, if the obtained trajectory violates the kinematic constraints of the humanoid manipulator, that is:

\begin{array}{l} {\begin{array}{l} θ_{i} (t) \notin [θ_{i}^{min}, θ_{i}^{max}] \\ {\overset{\cdot}{θ}}_{i} (t) \notin [{\overset{\cdot}{θ}}_{i}^{min}, {\overset{\cdot}{θ}}_{i}^{max}] \\ {\overset{\cdot\cdot}{θ}}_{i} (t) \notin [{\overset{\cdot\cdot}{θ}}_{i}^{min}, {\overset{\cdot\cdot}{θ}}_{i}^{max}] \end{array} \\ \forall t \in [0, t_{f 1} + t_{f 2}], ​ ​ i = 1,2, \dots, n \end{array}

(16)

then the constraint handling technique should be employed to punish the fitness value of the corresponding individual in DE:

f (K) = C

(17)

where $C > 0$ is a significantly positive constant.

5. Simulation

To validate the performance of the method proposed in this paper, as shown in Figure 3, a humanoid robot with a 7-DOF manipulator for ping-pong playing is employed as example. The joint structure model of the 7-DOF manipulator is in line with the physiological characteristics of the human arm, such that it has a large working space[17]. A detailed introduction about ping-pong playing with the humanoid robot has been described in [4]. The physical model and joint model of this humanoid manipulator are respectively shown in Figure 4, where $\sum_{W}$ is the world coordinate system and $a_{1} \sim a_{7}$ are respectively the rotation directions of the shoulder, elbow and wrist joint.

Figure 3.

Diagram of a humanoid robot for ping-pong playing

Figure 4.

Physical model (left) and joint model (right) of the 7-DOF manipulator

According to the actual mechanism model of the humanoid manipulator shown in Figure 4, the shoulder width is $D = 0.14$ m, the length of the upper arm is $L 1 = 0.26$ m, the length of the lower arm is $L 2 = 0.26$ m and the length from the wrist centre to the racket centre is $L 3 = 0.14$ m. In addition, the position range, the maximum velocity and the maximum acceleration of each joint are respectively listed in Tables 1 to 3:

Table 1.

Joint range of a 7-DOF humanoid manipulator (°)

Range	θ ₁	θ ₂	θ ₃	θ ₄	θ ₅	θ ₆	θ ₇
Upper	−126	−133	−180	−60	−180	−80	−42
Lower	90	15	90	120	180	80	85

Table 2.

Maximum joint velocity of a 7-DOF humanoid manipulator (rad/s)

${\dot{θ}}_{1}$	${\dot{θ}}_{2}$	${\dot{θ}}_{3}$	${\dot{θ}}_{4}$	${\dot{θ}}_{5}$	${\dot{θ}}_{6}$	${\dot{θ}}_{7}$
13	18	12	20	11	10	4.5

Table 3.

Maximum joint acceleration of a 7-DOF humanoid manipulator (rad/s²)

${\overset{\cdot\cdot}{θ}}_{1}$	${\overset{\cdot\cdot}{θ}}_{2}$	${\overset{\cdot\cdot}{θ}}_{3}$	${\overset{\cdot\cdot}{θ}}_{4}$	${\overset{\cdot\cdot}{θ}}_{5}$	${\overset{\cdot\cdot}{θ}}_{6}$	${\overset{\cdot\cdot}{θ}}_{7}$
200	150	300	200	120	200	100

According to the joint model of the 7-DOF manipulator in Figure 4, the unit vectors of the seven joint axes' directions $a_{1} \sim a_{7}$ are represented by the following formula[18, 19]:

{\begin{array}{l} a_{1} = (0,1,0) ​ a_{2} = (1,0,0) \\ a_{3} = (0,0,1) a_{4} = (1,0,0) \\ a_{5} = (0,0,1) a_{6} = (1,0,0) \\ a_{7} = (0,1,0) \end{array}

(18)

When defining the joint variables of the humanoid manipulator as a $7 \times 1$ vector $θ = {(θ_{1}, θ_{2}, θ_{3}, θ_{4}, θ_{5}, θ_{6}, θ_{7})}^{T}$ , if the neck orientation $R_{0}$ is equal to $E$ , then the position and orientation $(p_{j}, R_{j}) (i = 1,2, \dots,7)$ of each connecting rod is:

{\begin{array}{l} p_{j} = p_{i} + R_{i} b_{j} \\ R_{j} = R_{i} R_{a_{j}} (θ_{j}) \end{array}

(19)

where $p_{i}$ and $R_{i}$ are respectively the absolute position and orientation of the mother connecting rod in the world coordinate system, $a_{j}$ , $b_{j}$ are respectively the unit vector of axis directions and the origin coordinate in the coordinate system of the mother connecting rod, and $R_{a_{j}} (θ_{j})$ is the rotation matrix when the axis vector $a_{j}$ turns around θ_j radians. The rotation matrix can be calculated using the following Rodrigues' formula[4]:

\begin{array}{l} R_{a_{j}} (q_{j}) = E + {\hat{a}}_{j} sin q_{j} + {\hat{a}}_{j}^{2} (1 - cos q_{j}) \\ \hat{a} = {[\begin{array}{l} a_{x} \\ a_{y} \\ a_{z} \end{array}]}^{^} = [\begin{matrix} 0 & - a_{z} & a_{y} \\ a_{z} & 0 & - a_{x} \\ - a_{y} & a_{x} & 0 \end{matrix}] \end{array}

(20)

where $E$ is the unit matrix.

The movement of the humanoid manipulator for ping-pong playing includes the operation process and the return process. If the execution time of these two processes are $t_{f 1} = 0.3$ and $t_{f 2} = 0.5$ respectively, and the initial position at the zero moment and the target operation position at the $t_{f 1}$ moment of the humanoid manipulator are respectively given by:

\begin{array}{l} θ_{0} = [0 .0200 - 0 .5909 - 0.8551 1 .5930 \\ 1 .5708   0 .8441 - 0.5763] r a d \\ θ_{f} = [- 0 .3438 - 0 .3220 - 0.8813 1 .4156 \\ 1 .2132   0 .9866 - 0.1244] r a d \end{array}

(21)

In light of the forward kinematic model of this humanoid manipulator, the position and orientation of the end effector can be calculated according to the given joint vectors. Through simulation, the rod configuration of the humanoid manipulator at the initial moment (left) and at the target $t_{f 1}$ moment (right) are respectively shown in Figure 5.

Figure 5.

The rod configuration of the manipulator at the initial moment (left) and the target operation moment (right)

In addition, it can be assumed that the boundary conditions of joint velocity are given by:

\begin{array}{l} {\overset{\cdot}{θ}}_{0} = [0 0 0 0 0 0 0] r a d / s \\ {\overset{\cdot}{θ}}_{f} = [- 4 .7659   2 .1737 - 0 .1037 - 6.7378 \\ 0 .7667   2 .3225 - 2.0398] r a d / s \end{array}

(22)

In other words, the movement speed of the racket $V$ at the $t_{f 1}$ moment can be calculated to result in $V = [1.5 0.00 0.00]$ rad/s using the Jacobian matrix calculation. It is necessary to consider the influence on the balance of the humanoid robot resulting from the high-speed movement of the manipulator, since the humanoid biped robot tends to tip over.

Wang et al. [3] have concluded that smooth MATP is good for reducing disturbance. In order to obtain an optimum trajectory with minimum-acceleration and continuous acceleration properties, the DE method is adopted so that the optimization variables of the trajectory can be determined, i.e., the joint acceleration vector ${\overset{\cdot\cdot}{θ}}_{f}$ at the $t_{f 1}$ moment. As for the parameter setting in DE, the scaling factor and crossover probability of the DE strategy are $F = 0.75$ , $C R = 0.90$ , respectively, while the optimization variable dimension, i.e., the DOF of the manipulator, is $n = 7$ , and the range limitation of each variable is shown in Table 3. In addition, the population size of DE is set as $m = 30$ and the maximum evolutionary iteration is set as $T = 100$ . Furthermore, the fitness function of DE is defined as formula (15); moreover, according to the influence degree of each joint for the impact on the humanoid body, the weight parameters of the joint are respectively set as $w_{1} = 1.5$ , $w_{2} = 1.0$ , $w_{3} = 1.0$ , $w_{4} = 1.5$ , $w_{5} = 1.0$ , $w_{6} = 1.0$ and $w_{7} = 1.0$ . If the obtained trajectory violates the kinematic constraints, the penalty coefficient $C {= 10}^{4}$ is given to the fitness value of the corresponding individual. Through iteration optimization, the optimal joint acceleration vector ${\overset{\cdot\cdot}{θ}}_{f}$ at the $t_{f 1}$ moment can be obtained, as described here:

\begin{array}{l} {\overset{\cdot\cdot}{θ}}_{f} = [{\overset{\cdot\cdot}{θ}}_{1 f}, {\overset{\cdot\cdot}{θ}}_{2 f}, {\overset{\cdot\cdot}{θ}}_{3 f}, {\overset{\cdot\cdot}{θ}}_{4 f}, {\overset{\cdot\cdot}{θ}}_{5 f}, {\overset{\cdot\cdot}{θ}}_{6 f}, {\overset{\cdot\cdot}{θ}}_{7 f}] \\ = [13 .9239 - 15.6954 1 .2718 - 10.8801 \\ ​ ​ 31 .5082 - 3 .7232 - 24.7603] r a d / s^{2} \end{array}

(23)

where the corresponding fitness value is $f (K) = 405.0361$ . According to the fitness value, it can be seen that the selected trajectory meets all kinematic constraints of the humanoid manipulator. Figure 6 illustrates the fitness evolutionary process of DE. Meanwhile, Figure 6 shows that DE only needs about 50 iterations to reach convergence, with the convergence speed being very rapid.

Figure 6.

Fitness evolutionary process of DE

Figure 7 shows the position, velocity and acceleration versus the time curve of each joint for the selected optimum trajectory. From this figure, it can be seen that the acceleration trajectory is continuous everywhere and without discontinuous points, which is beneficial to the balance and stability of the humanoid body. In addition, the maximum acceleration $max | {\overset{\cdot\cdot}{θ}}_{i} (t) |$ of each joint can be determined, that is:

Figure 7.

Position, velocity and acceleration versus the time curve of each joint for the 7-DOF manipulator

\begin{array}{l} {\overset{\cdot\cdot}{θ}}_{max} = [max | {\overset{\cdot\cdot}{θ}}_{1} (t) |, \dots, max | {\overset{\cdot\cdot}{θ}}_{i} (t) |, \dots, max | {\overset{\cdot\cdot}{θ}}_{7} (t) |] \\ = [60.7352 23 .3161 1 .2734 100 .2489 \\ ​ ​ 31.5082 31 .1732 76 .2890] r a d / s^{2} \\ ​ t \in [0, t_{f 1} + t_{f 2}] ​ ​ ​ ​ ​ i = 1,2, \dots,7 \end{array}

(24)

In order to validate the performance of the proposed method in this paper, the method in the paper will be compared with the global minimum-jerk trajectory planning based on the DE method (MJTP-DE), whose main problem is to find the minimum-jerk trajectory within a determined time by using the DE algorithm. The jerk minimization problem during a determined time has been copiously studied[20 –22]. Huang et al.[20] used a genetic algorithm to search the optimal joint inter-knot parameters in order to realize the minimum jerk; Piazzi and Visioli[21] presented a new approach to searching for the global minimum-jerk cubic spline joint trajectory of a robot manipulator using interval analysis; Gasparetto and Zanotto[22] used fifth order B-splines to ensure that the squared jerk of the resulting trajectory was regarded as an optimal objective function.

In the MJTP-DE method, the quartic polynomial is also used to interpolate the joint trajectory of the whole motion, while the fitness function of DE is transformed into the following formula (25) when the trajectory satisfies all kinematic constraints of the humanoid manipulator.

\begin{array}{l} min f (K) = \sum_{i = 1}^{n} w_{i} \cdot max | {\overset{ṫ}{θ}}_{i} (t) | \\ t \in [0, t_{f 1} + t_{f 2}] ​ ​ ​ ​ i = 1,2, \dots, n \end{array}

(25)

where $| {\overset{ṫ}{θ}}_{i} (t) |$ is the jerk trajectory of the joint i and w_i represents the weight parameters, whose values are set as the aforementioned parameter values. Otherwise, the punishment measure $f (K) {= 10}^{4}$ is assigned to the fitness value when the trajectory breaks the kinematic constraints. In addition, the population size and the maximum evolutionary iterations, and other parameters, of the DE operator in the MJTP-DE method are set as the same as those in the aforementioned method in this paper. For the same assignment shown in formulae (21) to (22), through the MJTP-DE method, the joint accelerations at the $t_{f 1}$ moment are determined by:

\begin{array}{l} {\overset{\cdot\cdot}{θ}}_{f} = [{\overset{\cdot\cdot}{θ}}_{1 f}, {\overset{\cdot\cdot}{θ}}_{2 f}, {\overset{\cdot\cdot}{θ}}_{3 f}, {\overset{\cdot\cdot}{θ}}_{4 f}, {\overset{\cdot\cdot}{θ}}_{5 f}, {\overset{\cdot\cdot}{θ}}_{6 f}, {\overset{\cdot\cdot}{θ}}_{7 f}] \\ = [- 7 .2745 - 3.6206 0 .7783 - 32.9488 \\ ​ ​ 28.3398 5 .8652 - 24.7563] r a d / s^{2} \end{array}

(26)

In turn, each of the joint maximum accelerations of the trajectory by the MJTP-DE method can be obtained. Table 4 lists the maximum acceleration comparison of each joint trajectory planned by these two algorithms, respectively. From this table, it can be seen that the seven maximum accelerations of the proposed method in this paper are all smaller than those of the MJTP-DE method. Taking the first joint as an example, as shown in Figure 8, the maximum acceleration generated by the proposed method is 60.7352 rad/ $s^{2}$ , while the corresponding maximum acceleration of the MJTP-DE method is 81.9277 rad/ $s^{2}$ , which is about 1.35 times the former. Although both of these two curves can implement the same task, which is the point-to-point trajectory planning between the initial position and the target operation position, the proposed method generates the minimum-acceleration trajectory with a limited jerk for the joint, while the MJTP-DE method produces the trajectory with a smaller jerk, but with larger acceleration. Therefore, the trajectory planned by the method in this paper can generate less disturbance to the humanoid robot than the case with the MJTP-DE method. The proposed trajectory planning method is more feasible and effective.

Figure 8.

The acceleration and jerk trajectory for the first joint, where a solid curve is generated by the proposed method, whose maximum acceleration is 60.7352 $r a d / s^{2}$ and whose maximum jerk is $j_{m a x}$ ; meanwhile, for the dashed curve that is produced by the MJTP-DE method, the maximum jerk is 0.6546 times that of $j_{m a x}$ and the maximum acceleration is 81.9277 $r a d / s^{2}$

Table 4.

Trajectory maximum acceleration comparison of each joint for the two algorithms

Joint	maximum acceleration $max \| {\overset{\cdot\cdot}{θ}}_{i} (t) \|$ (rad/s²)
Joint	Proposed method	MJTP-DE
1	60.7352	81.9277
2	23.3161	35.3710
3	1.2734	1.7669
4	100.2489	122.3176
5	31.5082	34.6742
6	31.1732	40.5752
7	76.2890	76.2930

The differential evolution has solved the trajectory optimization problem of the humanoid manipulator operating in a rapid and continuous reaction environment through an evolution iteration procedure, which spends a little amount of time to obtain the problem solution. For high real-time operation tasks, e.g., ping-pong or baseball playing, this method will not satisfy the corresponding demands. In practice, this optimization process based on DE should be executed offline, that is, the optimized parameters of the movement trajectory are determined through DE offline, after which the obtained trajectory can meet the high real-time performance requirements online.

6. Conclusion

A kind of trajectory optimization method based on DE for a humanoid manipulator in a rapid and continuous reaction environment is presented. In this method, the whole trajectory of the humanoid manipulation is divided into two segments, that is, the operation process and the return process, and a quartic polynomial used to describe each segment. By optimizing the joint acceleration at the target operation moment using DE, a trajectory with a minimum-acceleration and a continuous acceleration profile is obtained to reduce the impact on the humanoid body. Simulation results on a humanoid robot with the 7-DOF manipulator show the effectiveness of the proposed method.

Despite these promising results, there is still room for improving our work in several aspects. It can be seen in Figure 8 that, when the quartic polynomial is adopted to describe the trajectory in the joint space, the jerk curve produces a discontinuous point at the target operation moment, which causes certain wear on the robot and probably shortens its lifespan. How to solve the jerk discontinuous phenomenon is, therefore, worth further study. Furthermore, in fitness function (15), the selection of the weight parameter values w_i results in a great impact on the trajectory optimization result; as such, these values should be reasonably set. Instead of using a priori knowledge, specific appropriate values ought to be obtained through parameterers effect analysis on the quality of the optimized trajectory. Consequently, the effect analysis of these parameters w_i in (15) could be discussed in the future. In addition, based on the research work in this paper, through optimization of the manipulator's reactive force and torque on a humanoid body, as well as analysing the zero moment point of a humanoid robot etc., a minimum-torque (or force) trajectory planning method should be studied to further reduce the disturbance to the humanoid robot.

Footnotes

7. Acknowledgements

This article is a revised and expanded version of a paper entitled “Optimal Trajectory Planning with Minimum-acceleration for a Humanoid Manipulator by Using Differential Evolution”, which was presented at the 2015 International Conference on Climbing and Walking Robots, held in Hangzhou, Zhejiang Province, China, 6-9 September 2015. The authors would also like to thank Xiong Rong, Liu Yong and Zhu Qiu-Guo from the Institute of Cyber-Systems and Control, Zhejiang University for providing the humanoid robot for this research work. This work is supported by the National Natural Science Foundation of China (Grant No. 61273340, 61203367).

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