Smooth trajectory generation is crucial for high performance control of actuators in industrial robots. In this article, we implement a simple jerk-limited time-optimal point-to-point reference trajectories for position control of electric motor based actuators. First, we briefly review the jerk-limited reference trajectory generation schemes. Then we simplify an existing result under the assumption that the initial and the final speeds of each reference trajectory are zero. Experimental results are provided to show the validity of the simplified algorithm.
Industrial robots require smooth reference trajectories to achieve high performance in motion control of their actuators [1]. In the motion control of electric motor based actuators, a reference trajectory that is not sufficiently smooth might cause undesirable phenomena such as degradation of control performance, mechanical vibration, over-current of drives, etc. The necessity for such smooth reference trajectories arise from the physical constraints imposed on electric motors and drives [2]. First, the acceleration trajectory of an actuator is essentially continuous, which implies the necessity for the continuity of the reference acceleration trajectory. To see the continuity of the acceleration trajectory, recall that the acceleration of the actuator is proportional to the stator current. Since the stator current is continuous, i.e., it cannot change abruptly because of the stator inductance, it is clear that the acceleration trajectory is always continuous. This constraint can be expressed as the boundedness of jerk, which is the instantaneous change rate of the acceleration. Second, the acceleration is bounded. This is due to the absolute rating of the electric motor and drive consisting of the actuator. Third, the speed of the actuator is bounded by the properties of the electric motor and drive. This is related to the electro-motive force generated by the electric motor and the DC link voltage of the drive. Thus, when planning the reference position trajectories, such physical constraints needs to be considered for the high performance motion control of the actuator.
Considering such physical constraints, trajectory generation schemes have been proposed [1, 3–6]. Among the existing schemes, polynomial interpolation based schemes have attracted research interests [3, 4]. Though smooth trajectories can be obtained based on high-order polynomial interpolation methods, they cause heavy computational load, which makes it difficult to implement.
To overcome the problem of the heavy computation load, the authors of [3] proposed a smooth speed reference generation algorithm using a fifth-order polynomial interpolation. Based on the result in [3], jerk-limited time-optimal reference speed trajectories can be obtained. The authors of [4] presented a jerk-limited time-optimal reference position trajectories. Though the algorithm presented in [4] is efficient and easy to implement, its computational load is still too heavy to be implemented in cost-effective microcontrollers.
Based on the above observations, we implement a simple jerk-limited time-optimal reference speed trajectories. To make the generation algorithm simple enough to be implemented in a cost-effective microcontroller, we assume that the acceleration and speed values at the initial and final instances are zero. Due to this assumption, reference trajectories become symmetric, which allows us to reduce the computation load. This simplified algorithm is implemented in a cost-effective microcontroller.
The remainder of this article is organized as follows. First, we briefly review the polynomial based jerk-limited reference trajectory generation in Section 2. In Section 3, we present the algorithm for the generation of jerk-limited time-optimal reference position trajectory under the assumption that the acceleration and speed values at the initial and the final instances are all zero. Simulation and experiment results are provided in Sections 4 and 5. We conclude the paper in Section 6.
When generating reference speed trajectories, we need to consider the following conditions and constraints:
where p denotes the position (or angle) of the actuator, and
where V denotes the maximum speed, A and D denote the maximum acceleration and deceleration, respectively, and J denotes the maximum jerk. The conditions in (
A generation algorithm for jerk-limited time-optimal reference trajectories was presented in [4] assuming that a_{0} =a_{f} =0. By the jerk-limited time-optimal trajectory for given (
Based on the algorithm in [4], the jerk-limited time-optimal reference position trajectories for given (
where τ_{k} = t −t_{k}_{–1} and
It should be noted here that the generation algorithm in [4] requires considerable amount of computation to find the time parameters, t_{1}, . . . ,t_{7}, though the reference position trajectories are given by (
Since the computational load required by the algorithm in [4] is not ignorable when the algorithm is implemented in a cost-effective microcontroller, it is desirable to have a simplified version of the algorithm. To this end, we additionally assume that v_{0} = v_{f} = 0, which allows us to simplify the algorithm based on the symmetry of the reference trajectories. Though the simplified version of the algorithm is less general, it is expected that the algorithm can be used for low-cost robotic systems.
The conditions for the generation of jerk-limited time-optimal reference position trajectories are summarized as follows:
The physical constraints to be considered are as follows:
and
Further we assume that the acceleration rate and the deceleration rate are identical for simplicity, i.e., A = D.
Depending on whether the speed and the acceleration values are limited in (
We need not consider the case in which the reference speed reaches its maximum value while the reference acceleration does not. This case does not make sure the time-optimality of the reference trajectory. In the following, we present the jerk-limited time-optimal trajectory for each case.
For brevity, we define T_{k} =t_{k}–t_{k}_{–1}. Due to the conditions in and the assumption that A = D, the reference position trajectory has point symmetry with the center at (
Due to the assumption that A = D, we obtain T_{1} = T_{3} = T_{5} = T_{7}. Further the traveling period can be divided into the acceleration, constant speed, and deceleration periods in general. For brevity, we denote the periods by X_{1}, X_{2}, and X_{3}, respectively. Definitions for the periods are given as follows:
By the point symmetry, the traveling distance L by the reference position trajectory is given as
where v_{p} is the peak speed during the traveling. The peak speed v_{p} is given as
which will be clear in below. Note that the traveling distance L needs to be the given distance P = p(T) − p(0).
We first consider the case that both of the reference speed and acceleration reach their maximum values. That is the case that v(t) and a(t) have V and A at some instances. A typical jerk-limited time-optimal reference position trajectory is shown in Figure 1. As shown in Figure 1, the reference acceleration reaches its maximum value A in T_{2} interval and the reference speed reaches its maximum value V in T_{4}.
Since the reference speed reaches its maximum value V in T_{4}, the peak speed v_{p} of the trajectory can be obtained as
Further the acceleration period T_{1} is obtained as
because the reference acceleration reaches its maximum value A and the jerk needs to be its maximum value J during the acceleration period to make sure the time-optimality. From the condition that the reference speed at the instance t =t_{3} is V, we have
which leads us to
To make sure that L = P, we have
which allows us to obtain
From (
We next consider the case in which the reference acceleration reaches its maximum value A in the interval t_{1} ≤ t ≤ t_{2} while the reference speed does not reaches its maximum value. This case corresponds to the case in which the traveling distance is not sufficiently long and thus the position reaches its final value without allowing the speed to reach its maximum value. A typical example of reference trajectory for
Since the reference acceleration reaches its maximum value and the peak speed of the reference trajectory is lower than the maximum speed, we have the following condition:
where
Due to the fact that the reference acceleration reaches its maximum value at the instance t = t_{1}, we have
By solving the
where we considers that T_{2} is not negative. Based on (
We then consider the case in which both the reference speed and reference acceleration do not reach their maximum value during the traveling. This case corresponds to the case in which the traveling distance is too short to allow the reference speed and reference acceleration to reach their maximum value. A typical example of reference trajectory for
For brevity, we denote the peak acceleration by A_{p}. Further we express the peak acceleration as
where 0 < α < 1. Then we can obtain
The peak speed is given by
From the condition that P = L, we have
which leads us to
From (
In this section, we provide simulation results of the jerk-limited time-optimal trajectory generation algorithm. The parameters for the simulation are summarized as follows.
Maximum jerk: 5,000 degree/s^{2};
Maximum acceleration: 3,000 degree/s^{2};
Maximum deceleration: 3,000 degree/s^{2};
Maximum speed: 6,000 degree/s;
Figure 4 shows the reference trajectory for
Figure 5 shows the reference trajectory for
Figure 6 shows the reference trajectory for
In this section, we provide experimental result of the jerk-limited time-optimal trajectory generation algorithm. We first briefly introduce the experimental setup and then provide the experimental results.
For the experiment, we use a commercial 400 W class servo motor. We further design and manufacture a servo motor drive based on a relatively low-cost digital signal processor. The overall configuration of the motor drive is shown in Figure 7. A vector control based position controller was implemented in the motor drive [2, 7]. The overall experimental setup is shown in Figure 8.
The parameters for the simplified trajectory generation algorithm in the experiment are the same as those in the previous simulation. Figure 9, 10, and 11 show the reference position, speed, acceleration, and jerk trajectories for
We implemented a jerk-limited time-optimal trajectory generation algorithm assuming zero initial and final speeds. Considering the computational load imposed by the existing generation algorithm, we provided a simplified version of the algorithm by additionally assuming that the speed values at the initial and final instances are zero. This additional assumption allows us to reduce the computational load of the trajectory generation algorithm. We provided simulation and experimental results to show the validity of the simplified algorithm.
This work was conducted within the project “Free Piston Engine Linear Generator for CHP” (No. EO170024) at the Korea Institute of Industrial Technology.
No potential conflict of interest relevant to this article was reported.
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Example of reference trajectory for
Example of reference trajectory for
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Simulation result for
Configuration of the servo motor drive.
Overall experimental setup.
Experimental result for
Experimental result for
Experimental result for