An adaptive velocity planning method for multi-DOF robot manipulators

Abstract

Multi-degrees of freedom robot manipulators are widely used in the manufacturing areas. Trajectory planning of the end effector is the most important while the hardest task for the use of the manipulators. An adaptive velocity planning method for multi-degrees of freedom robots with a predefined path is proposed in this article, which is based on interpolation technology. A resolved motion acceleration control strategy based on a fixed-distance movement is proposed, along with an adaptive time interpolation correction algorithm to achieve the speed planning for multi-degrees of freedom manipulators. Besides the accuracy and efficiency, more attentions were paid on the velocity control at turning points along the path, so that a transient vibration and a transient overshoot can be avoided, leading to a stable movement of the robot. A fold line trajectory and a circular-line connected trajectory were implemented on a 6-degrees of freedom robot manipulator. Simulations and experimental results show that the proposed method is efficient and effective.

Introduction

Multi-degrees of freedom (DOFs) robot manipulators are widely used in the manufacturing areas for their high flexibility of movement. With this flexibility, they are actually a new generation of tools that extend people’s physical and intellectual ability. The flexibility comes from the multi-DOF of the manipulator, which also brings many problems such as the solution of the inverse kinematic, the transient vibration and transient overshoot caused by the high-speed motion along complex traces, the movement accuracy and repeatability precision and so on. In order to improve the motion performances of multi-DOF robots, a lot of studies have been made in the literature in the field of robot kinematics, differential motions and velocities planning, dynamics and forces analysis, trajectory planning, motion control systems design, actuators and drive systems design and so on. In this article, we mainly concentrate on the motion control for multi-DOF robots.

The aim of motion control is to accomplish smooth and stable movement of the robot arms, which mainly relies on proper velocity and trajectory planning under the constraints of dynamics and forces. A fine movement for multi-DOF robots means that the end effector of the robots should move fast with high accuracy and high efficiency under the force and mechanics constraints.

Since the joints of a multi-DOF robot are linked one after another, the motion of them is dependent, so the motion control of the joints should be considered jointly. A friendly approach should achieve a multi-DOF coordinating movement without taking the risk of overshoot. Nowadays, without considering the design for mechanical driving devices, the motion control strategies for multi-DOF robots can mainly be divided into three classes, that is, the resolved motion rate control (RMRC), the resolved motion force control (RMFC) and the resolved motion acceleration control (RMAC).

The RMRC is a control strategy which changes the velocity of each joint during the whole motion process. According to this method, we can control each of the joint servos separately, but a cusp point in the speed curve will bring an abrupt change of the force and acceleration which will lead to a transient vibration and overshoot of the arm. Some researchers used smoothing techniques to adjust the motion; for example, Bruhm et al. ¹ proposed a fine move model by a filter to get the whole motion smoothly.

The RMFC aims to control the motion force for each joint. Kawamura and Fukao ² used the torque patterns obtained through learning control as the inputs and then another desired torque with a different timescale worked on the robot motion. The main advantage is that it has a compensation ability for the structure change of the robot, the gravity of each connecting rods and the inter friction. But for a random mechanical model, the motion planning for RMFC is hard because there are many types of forces in a multi-DOF system, and the mass of every component of the robot should also be considered in this strategy.

The RMAC is a strategy controlling the acceleration of the robots. Since the acceleration is the differential of the velocity and it relates to the force according to Newton’s second law, we can regard the acceleration as the linkage of the velocity and the force. A smooth acceleration control can provide a smooth motion of the robot. For multi-DOF robots, the control of the velocity and position of the end effector of a multi-DOF robot is usually achieved by servo motors and other drivers, and the control of the acceleration can also be achieved with them. A smooth RMAC strategy can enhance the robot’s efficacy and remove the sick phenomenon such as transient vibration and transient overshoot. The less transient vibration and transient overshoot the robot has, the higher speed and higher precision the motion will be.

The rapid progress of effective adaptive controllers has made a significant step forward of versatile applications of high-speed and high-precision robots. Compared with traditional closed-loop feedback control algorithms, a closed-loop feedback control with adaptive adjustment has illustrated that it is more effective and stable on the velocity, portability and control precision for robot motion control. ³ An adaptive trajectory planning strategy is powerful and robust to parameter uncertainties for robot manipulator control. Sawada and Itamiya ⁴ proposed a special adaptation control law, which shows superiority on the view point of adjusting the transient response of trajectory error.

Lotz et al ⁵ used experimental evaluation of identification methods to control the forces adaptively for industrial robotics. Tomimura et al. ⁶ proposed a smooth projection adaptation law in order to satisfy the constraints. Hernandez-Martinez et al. ⁷ proposed a formation tracking method based on approximate velocity. According to their method, a path matching can be utilized to construct a rigid formation, and the group trajectory tracking of robots can be achieved if adequate knowledge about positions and velocities of the robots is known. The smoothness of the trajectory is the main concern of these works.

The interpolation technology ⁸ has been widely used in precision lathes and other computer numerical control (CNC) systems. Interpolation can be done with high speed and high precision, so it has also been applied to multi-DOF robots. ⁹ Campos et al. proposed a robot trajectory planning approach for multiple three-dimensional (3D) moving objects interception based on polynomial interpolation. ¹⁰ Hirasawa et al. ¹¹ proposed a time-based interpolation control for robots. Wangyong and Li ¹² gave an interpolation-based time prediction to CNC systems. In our previous work, we also proposed a fixed-distance trajectory planning algorithm for 6-DOF robots based on Cartesian coordinates. ¹³ The key components to the interpolation operation are the interpolation step size and the interpolation time, which affect the precision and the speed of motion control. In this article, we extend the method in the study by Gao et al. ¹³ with a new adaptive velocity planning strategy based on the interpolation time. Some researchers have utilized time-based interpolation control in the literature.

As in the study by Gao et al., ¹³ a fixed-distance algorithm is also utilized for the trajectory generation in the Cartesian coordinate space to improve the positioning precision. The aim of the adaptive velocity planning strategy is to make the robot’s joints move stably and smoothly, especially at the connecting points of two adjacent sub-tracks.

The main contributions of this article are as follows. Firstly, an RMAC strategy based on a fixed-distance movement is proposed. Secondly, an adaptive velocity planning based on RMAC is proposed to achieve a high speed, accurate and stable trajectory planning for multi-DOF robots.

The remainder of this article is organized as follows. In ‘Brief introduction to the multi-DOF manipulator 3D modelling’ section, the 3D modelling of multi-DOF robots is reviewed and introduced. The proposed adaptive trajectory planning method is given in detail in ‘The adaptive RMAC control strategy’ section, especially for the velocity adjustment strategy. Both simulations and experimental results are shown in ‘Experimental verification’ section. The verification platform with our designed control hardware system is also described in this part. Finally, a brief summary is given in ‘The implemented hardware platform’ section.

Brief introduction to the multi-DOF manipulator 3D modelling

As we did in our previous work, ¹³ we still concentrate on the 6-DOF robot manipulator control in this article, and the same 6-DOF robot arm is used, which is made by Guangzhou Yardway Industrial Robots Ltd, China, as shown in Figure 1. The same link coordinate system for the manipulator using the Denavit–Hartenberg (D-H) representation is also established, as shown in Figure 2. With this D-H coordinate system established for each link, a homogeneous transformation matrix A_i, as given in formula (1), can be easily developed relating the ith coordinate frame to the (i-1)th coordinate frame. ¹⁴

Figure 1.

The utilized manipulator.

Figure 2.

The established link coordinate system.

In equation (1), Cθ and Sθ are used as the abbreviations to represent cosθ and sinθ, respectively, as in the literature, and similar meanings are with the

${\mathit{A}}_{\mathit{i}}=\left[\begin{array}{cc}\begin{array}{cc}\begin{array}{c}\begin{array}{c}C{\mathit{\theta }}_{\mathit{i}}\\ S{\mathit{\theta }}_{\mathit{i}}\end{array}\\ \begin{array}{c}0\\ 0\end{array}\end{array}& \begin{array}{c}\begin{array}{c}-S{\mathit{\theta }}_{\mathit{i}}C{\mathit{\alpha }}_{\mathit{i}}\\ C{\mathit{\theta }}_{\mathit{i}}C{\mathit{\alpha }}_{\mathit{i}}\end{array}\\ \begin{array}{c}S{\mathit{a}}_{\mathit{i}}\\ 0\end{array}\end{array}\end{array}& \begin{array}{cc}\begin{array}{c}\begin{array}{c}S{\mathit{\theta }}_{\mathit{i}}S{\mathit{\alpha }}_{\mathit{i}}\\ -C{\mathit{\theta }}_{\mathit{i}}S{\mathit{\alpha }}_{\mathit{i}}\end{array}\\ \begin{array}{c}C{\mathit{\alpha }}_{\mathit{i}}\\ 0\end{array}\end{array}& \begin{array}{c}\begin{array}{c}{\mathit{a}}_{\mathit{i}}C{\mathit{\theta }}_{\mathit{i}}\\ {\mathit{\alpha }}_{\mathit{i}}S{\mathit{\theta }}_{\mathit{i}}\end{array}\\ \begin{array}{c}{\mathit{d}}_{\mathit{i}}\\ 1\end{array}\end{array}\end{array}\end{array}\right]$ 1

notation of Cα and Sα too. The parameters for the six arms of the utilized robot in the D-H representation are shown in Table 1.

Table 1.

Arm link coordinate parameters.

Joint	θi	di (m)	ai (m)	ai	Joint range
1	θ1 + 90	0	0.175	−90	−180 to 180
2	θ2 + 90	0	0.6	0	−80 to 80
3	θ 3	0	0.11	−90	−210 to 70
4	θ 4	0.6	0	90	−180 to 180
5	θ5 + 90	0	0	90	−60 to 180
6	θ6 − 90	−0.14	0	0	−180 to 180

The pose of the end effector, which is represented by the matrix ${}_{ }{}^{0}T$ _H as shown in equation (2), can be calculated by multiplying all the transformation matrices A₁ to A₆.

${\mathrm{}}^0{\mathit{T}}_{\mathit{H}}=\left[\begin{array}{cc}\mathit{R}& \mathit{P}\\ 0& 1\end{array}\right]=\left[\begin{array}{cc}\begin{array}{cc}\begin{array}{c}{\mathit{n}}_{\mathit{x}}\\ {\mathit{n}}_{\mathit{y}}\end{array}& \begin{array}{c}{\mathit{o}}_{\mathit{x}}\\ {\mathit{o}}_{\mathit{y}}\end{array}\end{array}& \begin{array}{cc}\begin{array}{c}{\mathit{a}}_{\mathit{x}}\\ {\mathit{a}}_{\mathit{y}}\end{array}& \begin{array}{c}{\mathit{p}}_{\mathit{x}}\\ {\mathit{p}}_{\mathit{y}}\end{array}\end{array}\\ \begin{array}{cc}\begin{array}{c}{\mathit{n}}_{\mathit{z}}\\ 0\end{array}& \begin{array}{c}{\mathit{o}}_{\mathit{z}}\\ 0\end{array}\end{array}& \begin{array}{cc}\begin{array}{c}{\mathit{a}}_{\mathit{z}}\\ 0\end{array}& \begin{array}{c}{\mathit{p}}_{\mathit{z}}\\ 1\end{array}\end{array}\end{array}\right]={\displaystyle \prod }_{\mathit{i}=1}^6{\mathit{A}}_{\mathit{i}}$ 2

The objective of trajectory planning is to obtain an optimal solution to ${}_{ }{}^{0}T$ _H at each position along the path. Since all α_i s and d_i s are known and fixed as given in Table 1, we only need to find the optimal values for θ₁, θ₂,…, and θ₆ in order to obtain an optimal ${}_{ }{}^{0}T$ _H . In this article, we use the following ‘shortest stroke’ criterion ¹⁵ for trajectory planning, as we did in our last work. ¹³

$\underset{\mathit{\theta }}{\min }\Delta ={\displaystyle \sum }_{\mathit{i}=1}^6\left|{\mathit{\theta }}_{\mathit{i}\mathit{c}}-{\mathit{\theta }}_{\mathit{i}\mathit{l}}\right|$ 3

In equation (3), the θ_ic and θ_il are the current joint angle and the last joint angle for the ith joint, respectively.

The adaptive RMAC control strategy

In this article, we suppose that the path of the robot end effector is known beforehand and fixed. Our aim of trajectory planning is to let the robot move as fast and stably as it can. We utilize the same fixed-distance interpolation method as in our last article ¹³ for trajectory planning. Suppose the fixed distance for the interpolation, that is, the interpolation step size, is ΔS. Later in our implementation, we utilize a step size of 1 mm. If one needs a more accurate trajectory planning, a smaller step size can be utilized.

The foundation for the trajectory planning

When we consider each of the link of the manipulator as a rigid body, the forces required for accelerating and deaccelerating the links are a function of the desired acceleration and the mass distribution of the links. We can then utilize the Newton–Euler equations ¹⁶ to evaluate symbolically for any manipulator, and a dynamic equation can then be yielded as written in the following form ^17,18

$\mathit{\tau }=\mathit{M}\left(\Theta \right)\ddot{\Theta }+\mathit{V}\left(\Theta ,\dot{\Theta }\right)+\mathit{G}\left(\Theta \right)$ 4

where τ is the joint toque, Θ is a 6 × 1 vector of the joint angles, M (Θ) is the 6 × 6 mass matrix of the manipulator, $\mathit{V}\left(\Theta ,\dot{\Theta }\right)$ is a 6 × 1 vector of centrifugal and Coriolis terms and G (Θ) is a 6 × 1 vector of gravity terms. As depicted by the notation, each element of M (Θ) and G (Θ) is a complex function that depends on the positions of all the joints of the manipulator, and each element of $\mathit{V}\left(\Theta ,\dot{\Theta }\right)$ is a complex function of both Θ and $\dot{\Theta }$ .

On the other hand, since a fixed-distance strategy is utilized and the step size in this strategy, ΔS , is very small, Θ can be regarded as having no change for successive steps, and $\dot{\Theta }$ also changes very little. So, in this article, we simplify equation (4) and suppose that M (Θ), G (Θ) and $\mathit{V}\left(\Theta ,\dot{\Theta }\right)$ have no change for successive steps. According to this simplification, the change of the torque is only related to the angular acceleration, that is, the change of the angular velocity. So a discontinuity of the angular velocity will lead to a cusp in the torque, which will then cause motion vibration or transient overshoot of the robot. ¹⁷ The motion vibration and transient overshoot will bring such problems as decreasing the trajectory positioning accuracy, or increasing the mechanical losses of the robot, so a smooth acceleration control based on fixed-distance movement is friendly to us to achieve the stable motion. Zhu et al. ¹⁸ proposed an adaptive torque and position control for a three-legged robot which showed a meaningful adaptive control example for multi-DOF systems. But the complexity of the model should increase much if the number of DOFs increases which makes it hard for real applications.

The main constraint the robot should obey is that the torque τ for a single link should not exceed its maximum value, which can be simplified to the following simple equation according to the above analysis

$0\le \mathit{\alpha }\le {\mathit{\alpha }}^{\text{max}}$ 5

Actually this is the core idea for the RMAC strategy. In this article, the maximum acceleration along the path is set to be linearly related to the total length L of the path, that is

${\mathit{\alpha }}^{\mathbf{m}\mathbf{a}\mathbf{x}}=\mathit{k}\mathit{L}+\mathit{c}$ 6

where k and c are parameters which should be determined by the step size ΔS and the maximum torque that the servo motor can output.

The x ,  y and z components of α ^max at the ith interpolation position, that is, ${\mathit{\alpha }}_{\mathit{i},\mathit{x}}^{\mathbf{m}\mathbf{a}\mathbf{x}}$ , ${\mathit{\alpha }}_{\mathit{i},\mathit{y}}^{\mathbf{m}\mathbf{a}\mathbf{x}}$ and ${\mathit{\alpha }}_{\mathit{i},\mathit{z}}^{\mathbf{m}\mathbf{a}\mathbf{x}}$ can then be computed as follows

$\{\begin{array}{c}{\mathit{\alpha }}_{\mathit{i},\mathit{x}}^{\mathbf{m}\mathbf{a}\mathbf{x}}={\mathit{\alpha }}^{\mathbf{m}\mathbf{a}\mathbf{x}}\times \frac{{\mathit{p}}_{{\mathit{x}}_{\mathit{i}+1}-}{\mathit{p}}_{{\mathit{x}}_{\mathit{i}}}}{\Delta \mathit{S}}\\ {\mathit{\alpha }}_{\mathit{i},\mathit{y}}^{\mathbf{m}\mathbf{a}\mathbf{x}}={\mathit{\alpha }}^{\mathbf{m}\mathbf{a}\mathbf{x}}\times \frac{{\mathit{p}}_{{\mathit{y}}_{\mathit{i}+1}-}{\mathit{p}}_{{\mathit{y}}_{\mathit{i}}}}{\Delta \mathit{S}}\\ {\mathit{\alpha }}_{\mathit{i},\mathit{z}}^{\mathbf{m}\mathbf{a}\mathbf{x}}={\mathit{\alpha }}^{\mathbf{m}\mathbf{a}\mathbf{x}}\times \frac{{\mathit{p}}_{{\mathit{z}}_{\mathit{i}+1}-}{\mathit{p}}_{{\mathit{z}}_{\mathit{i}}}}{\Delta \mathit{S}}\end{array}$ 7

For the strategy of RMAC, the velocity curve should be smooth from the start to the end of the whole motion process; for a single rough point in the velocity, curve will lead to an abrupt force change at that point. For this reason, we choose the cosine/sine function as the basic velocity function, not only for the smoothness of the velocity themselves but also for the smoothness of their derivatives and integrations. In this article, the utilized velocity function is then as follows

$\mathit{v}\left(\mathit{t}\right)=\frac{1}{2}{\mathit{V}}^{\text{max}}\times \left[1-\cos \left(\frac{2\times \mathit{\pi }}{\mathit{T}}\mathit{t}\right)\right],0\le \mathit{t}\le \mathit{T}$ 8

From equation (8), we can get

$\mathit{a}\left(\mathit{t}\right)={\mathit{V}}^{\mathbf{m}\mathbf{a}\mathbf{x}}\frac{\mathit{\pi }}{\mathit{T}}\mathbf{s}\mathbf{i}\mathbf{n}\left(\frac{2\mathit{\pi }}{\mathit{T}}\mathit{t}\right)$ 9

From equation (9), we know that

${\mathit{a}}^{\mathbf{\text{max}}}=\frac{\mathit{\pi }}{\mathit{T}}{\mathit{V}}^{\mathbf{\text{max}}}$ 10

On the other hand, by integrating equation (10), we can obtain the total length of the path as follows

$\mathit{L}=\frac{1}{2}{\mathit{V}}^{\mathbf{\text{max}}}\mathit{T}$ 11

From equations (10) and (11), we can calculate the maximum speed V ^max and the operation time T as follows

${\mathit{V}}^{\mathbf{\text{max}}}=\sqrt{\frac{2\mathit{L}{\mathit{a}}^{{ }^{\mathbf{\text{max}}}}}{\mathit{\pi }}}$ 12 $\mathit{T}=\sqrt{\frac{2\mathit{\pi }\mathit{L}}{{\mathit{a}}^{\mathbf{\text{max}}}}}$ 13

The total number of interpolation points n for the whole path can be obtained according to equation (14)

$\mathit{n}=\frac{\mathit{L}}{\Delta \mathit{S}}$ 14

The distance S_i from the start point of the path at the interpolation time t _i is then

${\mathit{S}}_{\mathit{i}}=\text{i}\Delta \mathit{S}=\underset{0}{\overset{{\mathit{t}}_{\mathit{i}}}{{\displaystyle \int }}}\frac{{\mathit{V}}^{\mathbf{\text{max}}}}{2}\times \left[1-\cos \left(\frac{2\times \mathit{\pi }\times \mathit{t}}{\mathit{T}}\right)\right]\mathit{d}\mathit{t},\mathit{i}=1,2,\mathrm{..}\mathit{n}$ 15

In equation (15), t_i is the root of the following equation

$\mathit{f}\left({\mathit{t}}_{\mathit{i}}\right)= \frac{2\times {\mathit{S}}_{\mathit{i}}}{{\mathit{V}}^{\mathbf{\text{max}}}}+\frac{\mathit{T}}{2\mathit{\pi }}\times \text{sin}\left(\frac{2\mathit{\pi }{\mathit{t}}_{\mathit{i}}}{\mathit{T}}\right)-{\mathit{t}}_{\mathit{i}}=0$ 16

Equation (16) is a transcendental function and it can be solved with the Newton downhill method. With the interpolation time t_i obtained, we can calculate the operation time Δ t = t _{i + 1} − t _i for the ith step size Δ S . Figure 3(a) gives an example of the velocity curve for the end effector according to the velocity model (8), and Figure 3(b) shows the corresponding changing of the operation time according to the running time obtained from equation (16). As seen from Figure 3, the operation time at the beginning and the end of the motion is much larger than that at the middle of the motion. Since we have a fixed-distance motion strategy, we can conclude that the end effector accelerates first until it reaches its maximum velocity and then it decelerates, which is a reasonable process for real motion.

Figure 3.

The speed curve for end effector and the interpolation time on multi-DOF robot system. (a) The speed curve. (b) The interpolation time.

The adaptive velocity planning procedures

As aforementioned, velocity planning at turning points along the path is the key to motion transient vibration and transient overshoot avoidance. With special attentions paid to these turning points, the proposed detailed adaptive velocity planning procedures are shown in Figure 4.

Figure 4.

The flow chart for the proposed adaptive velocity planning method.

A detailed description of the procedures is given as follows:

Step 1. Input the path information.

Step 2. Compute its total length.

Step 3. Build the initial velocity model according to equation (10), with the parameters obtained according to equations (8), (14) and (15).

Step 4. Solve the interpolation time t_i ( i = 1,2,…, n ) according to equation (18) and calculate the corresponding acceleration a_i according to equation (11). Obtain the maximum operation time Δ t _max (which is almost the operation time at the start position) and the minimum operation time Δ t _min (the velocity reaches its maximum value at this position). Δ t _max is highly related to the maximum torque output by the servo motor; a smaller value of Δ t _max requires a bigger maximum output torque.

Step 5. Check if all a_i s ( i = 1,2,…, n ) satisfy the constraint (7). If equation (7) is satisfied for all a_i s, the velocity planning is accomplished, go to step 6.

If constraint (7) is not satisfied for some interpolation point C, do the following steps:

Record v_c and v _{c + 1}, and set Δ _tc to obey the following equation

$\{\begin{array}{c}\frac{{\mathit{v}}_{\mathit{c}+1,\mathit{x}}-{\mathit{v}}_{\mathit{c},\mathit{x}}}{\Delta {\mathit{t}}_{\mathit{c}}}=\frac{\Delta {\mathit{v}}_{\mathit{c},\mathit{x}}}{\Delta {\mathit{t}}_{\mathit{c}}}\le {\mathit{a}}_{\mathit{x}}^{\mathbf{\text{max}}}\\ \frac{{\mathit{v}}_{\mathit{c}+1,\mathit{y}}-{\mathit{v}}_{\mathit{c},\mathit{y}}}{\Delta {\mathit{t}}_{\mathit{c}}}=\frac{\Delta {\mathit{v}}_{\mathit{c},\mathit{y}}}{\Delta {\mathit{t}}_{\mathit{c}}}\le {\mathit{a}}_{\mathit{y}}^{\mathbf{\text{max}}}\\ \frac{{\mathit{v}}_{\mathit{c}+1,\mathit{z}}-{\mathit{v}}_{\mathit{c},\mathit{z}}}{\Delta {\mathit{t}}_{\mathit{c}}}=\frac{\Delta {\mathit{v}}_{\mathit{c},\mathit{z}}}{\Delta {\mathit{t}}_{\mathit{c}}}\le {\mathit{a}}_{\mathit{z}}^{\mathbf{\text{max}}}\end{array}$ 17

So we have

$\Delta {\mathit{t}}_{\mathit{c}}=\text{max}\left\{\frac{ \Delta {\mathit{v}}_{\mathit{c},\mathit{x}}}{{\mathit{a}}_{\mathit{x}}^{\mathbf{\text{max}}}},\frac{ \Delta {\mathit{v}}_{\mathit{c},\mathit{y}}}{{\mathit{a}}_{\mathit{y}}^{\mathbf{\text{max}}}},\frac{ \Delta {\mathit{v}}_{\mathit{c},\mathit{z}}}{{\mathit{a}}_{\mathit{z}}^{\mathbf{\text{max}}}}\right\}$ 18

Resolve t _c , t _{c + 1}, V^max and T according to the following equations

$\{\begin{array}{c}\begin{array}{c}\Delta {\mathit{t}}_{\mathit{c}}={\mathit{t}}_{\mathit{c}+1}-{\mathit{t}}_{\mathit{c}}\\ \mathit{f}\left({\mathit{t}}_{\mathit{c}+1}\right)=\frac{2{\mathit{s}}_{\mathit{c}+1}}{{\mathit{V}}^{\mathbf{\text{max}}}}+\frac{\mathit{T}}{2\mathit{\pi }}\text{sin}\left(\frac{2\mathit{\pi }{\mathit{t}}_{\mathit{c}+1}}{\mathit{T}}\right)-{\mathit{t}}_{\mathit{c}+1}=0\\ \mathit{f}\left({\mathit{t}}_{\mathit{c}}\right)=\frac{2{\mathit{s}}_{\mathit{c}}}{{\mathit{V}}^{{ }^{\mathbf{\text{max}}}}}+\frac{\mathit{T}}{2\mathit{\pi }}\text{sin}\left(\frac{2\mathit{\pi }{\mathit{t}}_{\mathit{c}}}{\mathit{T}}\right)-{\mathit{t}}_{\mathit{c}}=0\end{array}\\ \begin{array}{c}\Delta \mathit{S}=\underset{{\mathit{t}}_{\mathit{c}}}{\overset{{\mathit{t}}_{\mathit{c}+1}}{{\displaystyle \int }}}\frac{{\mathit{V}}^{\mathbf{\text{max}}}}{2}\left(1-\cos \left(\frac{2\mathit{\pi }\mathit{t}}{\mathit{T}}\right)\right)\mathit{d}\mathit{t}\\ {\mathit{s}}_{c+1}=\left(c+1\right)\cdot \Delta S\\ {\mathit{s}}_c=C\cdot \Delta S\end{array}\end{array}$ 19

With the newly obtained V^max and T , rebuild the velocity model according to equation (10). Repeat step 4 to step 5 until the constraint (7) is satisfied.

Step 6. Feed the obtained operation time Δt_i = (i = 1,2,…, n − 1) to the controller.

From step 1 to step 4, we do an ordinary velocity planning without adaptiveness, the only consideration of these steps is the motion speed. And step 5 is actually the adaptive adjustment process for the velocity curve to make the motion smooth and stable further.

Figure 5(a) shows a typical changing of the operation time according to the running time based on our proposed adaptive velocity planning method when there is a turning point existing along the path (please refer to the real examples in ‘Experimental verification’ section). As seen from Figure 5(a), the obtained operation time increases very much so that the end effector moves much slowly at the turning point in order to avoid motion vibration and motion overshoot. Figure 5(b) is the correction result for the path which has a turning point.

Figure 5.

The adaptive velocity planning results for the end effector of a multi-DOF robot. (a) The adjusted interpolation time. (b)The adjusted velocity.

One thing worthy of mentioning is that the interpolation error may be ignored, but the dynamic accumulation error cannot be neglected. In order to provide a high positioning accuracy, a mechanism of feeding back the accumulation error and the interpolation error is better to be provided to revise the track.

Experimental verification

In this part, we show two examples of velocity planning for two different paths, respectively, based on the proposed method; one is a broken line path and the other is an-arc-joined-with-a-line path, as shown in Figure 6. To represent the orientations at each interpolation point, we utilize the quaternion representation method ¹⁹ which has been widely used in the 3D graphics animation field.

Figure 6.

The two paths used for experimental verification. (a) The broken line path. (b) The arc-line path.

The broken line path

The broken line path for the experimental verification is shown in Figure 6(a).

The obtained joint angle, angular velocity and angular acceleration for each of the joint according to the interpolation points for the ordinary velocity planning method and adaptive velocity planning method are shown in Figure 7. As seen from Figure 7(a), a discontinuous point exists at the 655th interpolation time for the acceleration curves, which brings motion transient vibration. The corresponding variations of the joint angles, angular velocities and angular accelerations according to the adaptive velocity planning method for the six joints are shown in Figure 7(b). As seen from Figure 7(b), the motion becomes much more smoothly. In the whole motion, the angular velocity curves are very smooth, and the angular acceleration curves are continuous. So no transient vibration or transient overshoot occurs during the whole motion process when it is tested for the real multi-DOF robot. The comparison between Figure 7(a) and 7(b) shows that the proposed adaptive velocity planning method is superior to the ordinary one.

Figure 7.

Quantities variations under different velocity planning methods for the broken line path. (a) Under the ordinary method. (b) Under the proposed adaptive method.

The proposed method was also tested on our implemented hardware system, a detailed description of which will be shown in the ‘The implemented hardware platform’ section. From the integrated development environment, Code Composer Studio (CCS), provided by Texas Instruments ( Boulevard Dallas, Texas, USA), we can obtain the path information, which was then fed to MATLAB to compute ${}_{ }{}^{0}T$ _H . We can also obtain the running clock cycles from CCS. With the CPU frequency and the interruption period known, the operation time can then be calculated. The obtained length and time quantities for the broken line path are shown in Table 2. The total length of the path is 828.00 mm, and the total number of interpolation points is 828. The total operation time for the ordinary velocity planning is 1.5584 s, while the total operation time for the adaptive velocity planning is 2.05562 s in our designed hardware platform, as shown in Table 2. An average error of 0.04578645 mm for each 1-mm interpolation distance is obtained for the ordinary method, while the value for the proposed adaptive method is only 0.00454 mm. For a comparison, we also list the values for Δ t_max , Δ t_min and Δ t_c in Table 2.

Table 2.

Actual operation quantities for the broken line path.

Test number Variables	First	Second	Third	Fourth	Fifth	Ideal parameter	Average error
The length of path (mm)	824.02	825.03	824.03	824.05	824.07	828.00	−3.76
Operation time (ms)	2046.385	2046.366	2046.347	2046.333	2046.360	2055.62	−9.2618
Δ tmax (ms)	88.37	88.36	88.37	88.40	88.34	88.38	−0.012
Δ tmin	1.101 ms	1.084 ms	1.081 ms	1.083ms	1.081ms	1.0944 ms	−0.0084 ms
Δtc (ms)	47.90	47.92	47.99	47.89	47.94	47.95	−0.022
Error at the end point (mm)	0.00745	0.00721	0.007754	0.007845	0.007536		0.007559

The an-arc-joined-with-a-line path

The an-arc-joined-with-A-line path for illustration is shown in Figure 6(b). The path information is as follows: the total length of the path is 58.10 cm and the number of interpolation points is 581. The path starts at a position (x: 29.80 mm, y: 10.00 mm, z: 21.27 mm) and ends at a position (x: 50.00 mm, y: 50.00 mm, z: 21.30 mm). The radius of the arc is 10.00 mm and it ends at a position (x: 20.00 mm, y: 19.99 mm, z: 21.27 mm). The obtained joint angle, angular velocity and angular acceleration for each of the joint according to the interpolation points are shown in Figure 8(a). As seen from Figure 8(a), a discontinuous (impulse) point exists at the 155th interpolation time for the acceleration curves, which also brings motion transient vibration.

Figure 8.

Quantities variations under different velocity planning methods for the arc-line path. (a) Under the ordinary method. (b) Under the proposed adaptive method.

With a comparison, the obtained variations of the joint angles, angular velocities and angular accelerations for the six joints for the adaptive velocity planning method are shown in Figure 8. As seen from Figure 8(b), the motion becomes much more smoothly compared with that in Figure 8(a). Again no discontinuous point appears in the whole motion. And no transient vibration or transient overshoot occurs during the whole motion process when it is tested for the real industrial robot. The measured parameters for this example are shown in Table 3. The total operation time for the ordinary velocity planning method is 1.7930 s, while the operation time for the adaptive velocity planning method is 2.1272 s. An average error of 0.0188 mm for each 1 mm interpolation distance is obtained for the ordinary method, and the value for the adaptive method is only 0.00183 mm.

Table 3.

Actual operation quantities for the arc-line path.

Test number Variables	First	Second	Third	Fourth	Fifth	Ideal parameter	Average error
The length of path	579.95 ms	584.51 ms	583.80 mm	578.13 mm	583.93 mm	581.00 mm	1.064 mm
Operation time (ms)	2131.444	2131.430	2131.424	2131.443	2131.438	2127.212	4.2238
Δt max (ms)	68.55	68.74	68.77	68.67	68.75	68.70	−0.004
Δt min	1.693 ms	1.694 ms	1.689 ms	1.695 mm	1.693 ms	1.691 ms	0.0018 ms
Δtc (ms)	68.55	68.74	68.77	68.67	68.75	68.70	−0.004
Error at the end point (mm)	0.00587	0.00657	0.00614	0.00624	0.00631		0.006226

Both examples show that our proposed adaptive velocity planning method can control the end effector move high efficiently with high accuracy, without any motion vibration and motion overshoot. Another advantage of the proposed method is its flexibility to the path, that is, velocity planning can be done adaptively for any known but fixed path. Another important thing worthy of mentioning is that positioning errors still exist for the proposed method. This positioning error is actually accumulated with the interpolation points, so the longer the path, the bigger is the error. Actually this is a common problem for the interpolation-based methods. Bartok and Vasarhelyi ²⁰ analysed a fuzzy interpolation method implemented on different platforms and then pointed out that the accumulation error needed attention. Bai et al. ²¹ proposed a novel fuzzy interpolation algorithm to deal with the position or orientation errors for industrial robot calibrations and manufacturing processes. In interpolation, accumulative errors will be the key. As in the literature, a feedback control system based on fuzzy interpolation may be a good solution to handle this problem.

The implemented hardware platform

In order to test the proposed method for real applications, we built an embedded hardware platform. The flowchart of the whole hardware system is shown in Figure 9(a). The whole system is composed of four parts, that is, the controller board, the vision board, the servo motors with drivers and the robot manipulator. The controller board runs the proposed algorithm to control the motion of the manipulator. The main components of the controller board are the digital signal processor (DSP) module and the field programmable gate array (FPGA) module. The DSP module is used for algorithm running, with a central processing unit (CPU) of OMAPL138, which is made by Texas Instruments Inc., which contains a DSP core of C674X and an ARM9 core. The DSP runs at a frequency of 456 MHz. In real applications, we utilize an interruption mode to collect the data, and the interruption works at a frequency of 10 MHz. The FPGA, EP3C16Q240C8 N, which is made by Altera (Shanghai, China), accepts the control data from the OMAPL138 module and then controls the robot manipulator to move along the desired trajectory with the help of the servo motors driven by the servo drivers. The vision board generates the start and end positions of the trajectory with certain computer vision algorithms. The designed motion control board is shown in Figure 9(b).

Figure 9.

The flowchart of the designed hardware platform and the motion board. (a) The structure of the hardware platform. (b) The designed motion control board.

Summary

In this article, an adaptive velocity planning method for known but fixed paths is proposed, based on a fixed-distance interpolation strategy. The core idea of the proposed method is based on the RMAC strategy. In order to have a smooth acceleration curve, we first build a cosine function velocity model with its parameters obtained according to the path length, then all the interpolation time for each interpolation distance are solved, the obtained acceleration values at each interpolation time are then checked with the constraint, and finally an recursive adaptive adjustment of the operation times is done for those interpolation points violating the constraint until all the accelerations obey the constraint. Simulations and experimentations show that our proposed adaptive RMAC-based velocity planning method is effective and efficient, providing a fast and accurate movement control for the end effector of industrial robots without motion transient vibration and transient overshoot.

The research can be further improved in several aspects. The algorithm of the solution to the inverse kinematics can be further optimized to improve the efficiency. A more reliable control system may also be needed in the future.