Using a trajectory-following controller provides several significant benefits, especially in systems requiring smooth, precise, and reliable movements.
Neyram Hemati, Ph.D.
Department of Mechanical Engineering, San Jose State University
Don Macleod, Ph.D.
CEO, Applied Motion Products
Model-based feedforward control is a predictive control strategy using a mathematical model of the system to predict the control input needed for a desired trajectory. Unlike feedback control, which reacts to errors, feedforward control anticipates disturbances or system dynamics and applies control actions so that unwanted effects are cancelled [1].
Trajectory control is typically composed of two parts: planning and tracking [1][2]. During the planning phase, the geometry of motion in terms of displacement, velocity, and acceleration as functions of time is formulated. During the tracking phase, advanced control schemes involving feedback and feedforward laws are used to keep the error between the reference trajectory and the actual trajectory as small as possible.
Here, we consider these techniques in motion control and robotics where the goal is to precisely guide a system (e.g., a robot arm, drone, vehicle) along a desired path or trajectory, often at high speed and with minimal error [1][2]. Some commercially available servo drives, such as the MDX+ integrated series from Applied Motion Products (AMP), include configurable velocity and acceleration feedforward parameters. These features allow for precise control in high-performance, dynamic environments by improving trajectory tracking, minimizing response lag, and enhancing overall system performance. This level of control is essential in high-speed, high-precision applications like robotics, CNC machining, semiconductor processing, autonomous systems, and advanced manufacturing.
Here, the focus is single-axis systems where the problem corresponding to the coordinated motion among multiple joints and links, e.g. multi degree-of-freedom robots, is not of concern. The trajectory control impact is in terms of improved tracking performance, response time, and reduced overshoot in high-speed applications like pick-and-place systems or laser cutting.
Feedforward control
Feedforward control is particularly useful in systems with nonlinear dynamics (e.g., robots, drones) where it’s used to effectively linearize the dynamics of the system. As special cases of feedforward control, there are model-based approaches where the control input is computed by inverting the dynamic model of the plant (e.g., torque required to follow a certain acceleration). For example, as a model-based approach, feedforward control strategies can help in terms of following time-varying trajectories, e.g. trapezoidal acceleration, as well as rejecting time-varying disturbances [3]. Feedforward control can also be adaptive in nature where the controller adapts to system changes over time, such as load variations. Most systems that use feedforward control also use feedback control laws for stability purposes and robustness.
As an example, feedforward control in robotics uses the dynamic model of the robot in conjunction with the desired trajectory to compute the control effort which would nominally cancel the targeted effects. To help achieve smooth motion, the desired reference acceleration is usually used as a feedforward term [3]. Also, for example, based on the desired state of the system, e.g. position, an attempt can be made to cancel the gravitational effects [1][2][3]. On the other hand, measured states can be used to provide feedback cancellation for gravity as well as velocity-dependent effects. However, cancelling the effect of gravity and velocity-dependent terms alone does not guarantee adequate performance of the system because of disturbances and model uncertainties. As a result, it’s desired if not necessary to provide feedback control. A PID feedback control law can be used to help meet the performance criteria [3][4].
Trajectory planning
For a motion control system to achieve smooth motion between the start and the end points, it’s necessary to have a position profile as function of time that is smooth. In addition, having a smooth velocity as well as acceleration further provide for smooth motion. The smoother the motion, the less the possibility for exciting vibration modes and the less cause for wear and damage to the mechanical components.
The task of trajectory planning is to formulate the desired time histories of position, velocity, and acceleration corresponding to a given move or task by the motion control system. For instance, a trajectory may be defined in terms of a cubic or a quintic polynomial describing position as a function of time [1][2]. More common are the so-called trapezoidal velocity and trapezoidal acceleration trajectories [3]. For the trapezoidal acceleration and trapezoidal velocity trajectories, an example is provided in Figure 1.
Trajectory control
In trajectory control, the reference trajectory—defined in terms of position, velocity, and acceleration over time—is provided to the control system. The system then uses feedback control laws to ensure that actual outputs, such as position and velocity, closely follow the reference trajectory with minimal tracking error. This is typically achieved using a cascade or nested-loop control structure [3], where PID controllers are commonly employed for both the inner and the outer position loop.
Feedforward trajectory control sample results
Applied Motion Products’ MDX+ series servo drives offer advanced support for feedforward control and trajectory tracking, making them well-suited for applications requiring high dynamic performance and accuracy [4]. Using Applied Motion Products’ Luna software [3], we defined a reference trapezoidal velocity trajectory for an Applied Motion Products MDX+ servo drive and executed the move with and without the reference velocity, and reference acceleration, as inputs to the nested-loop PID controller. Luna also provides a “Gravity Compensation” option [3].
The figure below shows a comparison of tracking error results with and without the use of reference velocity and acceleration as feedforward signals. The results demonstrate that including these feedforward terms significantly reduces the maximum tracking error. Additionally, the figure illustrates that by properly tuning the gains associated with the feedforward velocity and acceleration, the tracking error can be reduced to nearly zero.
Summary
As a model-based approach, feedforward control strategies can help in terms of following time-varying trajectories, e.g. trapezoidal acceleration, as well as rejecting time-varying disturbances. Using a trajectory tracking controller provides several significant benefits, especially in systems requiring smooth, precise, and reliable movements. This also results in improved mechanical longevity and reduced wear as well as enhanced tracking accuracy. Plus, using a smooth reference trajectory would help achieve a system with reduced noise and vibration issues.
Applied Motion Products
www.applied-motion.com
References
[1] J. J. Craig, Introduction to Robotics: Mechanics and Control, 3rd ed. Pearson-Prentice Hall, 2005.
[2] M. W. Spong, S. Hutchinson, and M. Vidyasagar, Robot Modeling and Control. Hoboken, NJ: John Wiley & Sons, 2005.
[3] Luna Software User Manual, Applied Motion Products, Morgan Hill, CA, 2025. [Online]. Available: https://www.applied-motion.com/s/software
[4] MDX+RC Series Hardware Manual, Applied Motion Products, Morgan Hill, CA, 2025.
Filed Under: Motion Control Tips
