Industrial robot high-order smooth trajectory generation method

Abstract

This invention relates to the field of robot motion control technology and provides a method for generating high-order smooth trajectories for industrial robots. The method includes: constructing a fifth-order polynomial trajectory planning model and establishing start and end state boundary conditions; adaptively tuning the running time based on physical limit constraints and discretizing it; constructing a linear matrix equation and analytically solving the polynomial coefficients to generate discretized trajectory instructions; and issuing the instructions to the underlying control system for closed-loop execution. This invention utilizes the high-order continuity of the fifth-order polynomial to achieve globally smooth and bounded jerk, completely avoiding abrupt jerk step jumps; it adopts a deterministic algebraic analytical computation architecture, avoiding complex piecewise logic and iterative solutions, significantly reducing computational overhead and timing fluctuations. This invention ensures high-order trajectory smoothness while meeting real-time requirements, effectively reducing mechanical soft impacts, and is suitable for the underlying servo control of multi-axis industrial robots.

Description

Technical Field

[0001] This invention belongs to the field of robot motion control technology, and in particular relates to a method for generating high-order smooth trajectories for industrial robots. Background Technology

[0002] With the increasing demands of intelligent manufacturing, industrial robots are rapidly evolving towards higher speeds and lighter weights. As a typical multi-body flexible system, robotic arms are susceptible to mechanical shocks and residual vibrations induced by discontinuous kinematic constraints when performing high-speed movements. These shocks not only prolong the system's settling time, but their long-term cumulative effects also accelerate the wear of mechanical components and shorten the equipment's service life. Therefore, researching smooth trajectory planning algorithms with high-order continuity to weaken excitation energy at the input end is of core engineering significance for improving the dynamic accuracy of industrial robots.

[0003] Currently, the smooth trajectory generation algorithms widely used in industry and academia mainly include low-order spline interpolation algorithms, S-shaped velocity planning and its variants, as well as optimization-based trajectory generation frameworks. Summary of the Invention

[0004] The purpose of this invention is to provide a method for generating high-order smooth trajectories for industrial robots, aiming to solve the problems existing in the background art.

[0005] This invention is implemented as follows: a method for generating high-order smooth trajectories for industrial robots, comprising the following steps:

[0006] Step 1: Trajectory Mathematical Modeling and Boundary Condition Establishment

[0007] A trajectory planning model based on fifth-order polynomials is constructed, and the angular displacement function and its kinematic derivative equations are established. The motion states at start and end times are defined, and global boundary conditions are established. (Start time...) and the end time The motion states are described by the initial pose motion state vector and the target pose motion state vector, respectively. Based on the kinematic conservation rules, a global boundary condition containing six independent constraints is established, and a boundary condition vector is constructed accordingly.

[0008] Step 2: Adaptive Tuning of Runtime

[0009] Based on physical limit constraint parameters, the global running time of the trajectory is adaptively tuned. By analyzing the kinematic extreme value distribution characteristics of the fifth-order polynomial trajectory in the entire time domain, the theoretical shortest running time satisfying each order of physical limit constraints is calculated. This theoretical shortest time is then rounded up and strictly aligned to an integer multiple of the discretized control interpolation period of the underlying controller to obtain the final discretized running time used for online solution. .

[0010] Step 3: Analytical solution of polynomial coefficients and generation of discrete trajectory

[0011] Substituting the global boundary conditions established in step one into the fifth-order polynomial and its derivative equations, a linear matrix equation is constructed. Substituting the discretized runtime obtained in step two, the explicit analytical solution of the fifth-order polynomial trajectory coefficients is directly calculated using the determined algebraic formula, thereby generating a discretized sequence of position, velocity, and acceleration trajectory commands.

[0012] Step 4: Closed-loop execution of trajectory commands

[0013] The discrete trajectory command sequence generated in step three is sent to the industrial robot's underlying control system for real-time closed-loop execution. After receiving the discrete trajectory command, the underlying control system uses a proportional-integral-derivative control law for servo tracking by the joint trajectory controller, and provides high-frequency feedback on the joint state to form a full closed-loop control.

[0014] The present invention provides a method for generating high-order smooth trajectories for industrial robots, which has the following beneficial effects:

[0015] Global trajectory realization using the high-order continuity properties of quintic polynomials It is continuous and avoids the sudden acceleration step change in the traditional S-curve algorithm, effectively weakening the energy of the high-frequency excitation source from the physical input end.

[0016] By employing a deterministic algebraic analytical computation architecture, complex nonlinear piecewise logic and transcendental equation iterations are avoided, significantly reducing the average execution time per step and the maximum single-step observation time and standard deviation of the time consumption. This provides a highly deterministic and low-time-consumption online instruction generation solution for resource-constrained embedded controllers.

[0017] By inputting a highly smooth trajectory and appropriately limiting kinematic extrema, the transient impact of end-effector inertia changes on the underlying servo controller is mitigated. When relying solely on standard proportional-integral-derivative closed-loop control and experiencing full-load perturbation, the dynamic tracking error is strictly limited to a very small range.

[0018] While significantly improving the smoothness of trajectory kinematics, it significantly reduces the complexity of algorithm execution, meets the real-time and timing stability requirements of high-frequency control cycles of industrial robots, and can be directly deployed in the underlying servo control system of multi-axis industrial robots. Attached Figure Description

[0019] Figure 1 This is a flowchart of the trajectory generation algorithm based on fifth-order polynomial interpolation of the present invention;

[0020] Figure 2 This is a diagram of the closed-loop control simulation system architecture of the present invention;

[0021] Figure 3 This is a comparison diagram of the position curves of the present invention and the prior art;

[0022] Figure 4 This is a comparison chart of the speed curves of the present invention and the prior art;

[0023] Figure 5 This is a comparison chart of the acceleration curves of the present invention and the prior art;

[0024] Figure 6 This is a comparison diagram of the acceleration frequency domain characteristics of the present invention and the prior art;

[0025] Figure 7 This is a diagram showing the shoulder joint tracking error under different load conditions according to the present invention. Detailed Implementation

[0026] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention.

[0027] The specific implementation of the present invention will be described in detail below with reference to specific embodiments.

[0028] Example 1

[0029] like Figure 1 As shown, this embodiment provides a method for generating high-order smooth trajectories for industrial robots, which is applied to the single-segment point-to-point motion control of a six-degree-of-freedom industrial robotic arm.

[0030] System kinematic variables and constraint definitions

[0031] First, the core kinematic variables and physical limit constraint parameters involved in the system are defined in a unified manner.

[0032] Trajectory function Used to characterize the angular displacement in the joint space of an industrial robot over time. The evolutionary law of change, its first derivative Let be the joint angular velocity, and be the second derivative. Let be the joint angular acceleration, and the third derivative. Add acceleration to the joints.

[0033] Physical limit constraints include: the maximum permissible speed V of the joint. max Maximum permissible acceleration A max and maximum permissible jerk J max These parameters are determined by the physical characteristics of the underlying servo motor.

[0034] Control interpolation period This represents the underlying timing interval for the low-level motion controller to perform a single discrete trajectory point calculation and issue communication commands. In this embodiment, the high-frequency interpolation period is set to 125 Hz, i.e. It takes 0.008 seconds.

[0035] Step 1: Trajectory Mathematical Modeling and Establishment of Global Boundary Conditions

[0036] To meet the requirements for smooth and flexible transitions between the start and end states of the robotic arm's joints, a trajectory planning fundamental mechanism model is constructed using fifth-order polynomials as basis functions. The angular displacement function and its kinematic derivative equations are established as follows:

[0037]

[0038] in, The coefficients are undetermined polynomials, and the running time for trajectory planning is set to... ,in To discretize the runtime.

[0039] As can be seen from the above mechanism model, the jerk characterizes the system's flexible impact strength. Directly transformed into something about time A continuous quadratic function. Since the fifth-degree polynomial lies on the open interval... The model inherently possesses infinite order continuous characteristics, which makes the jerk exhibit a globally smooth and bounded state, thus completely eliminating the abrupt jerk changes that exist in traditional low-order polynomial trajectories from a physical mechanism perspective.

[0040] For typical working conditions where industrial robots perform single-segment point-to-point motion, the starting time is... and the end time The motion state is subject to strict global constraints. The initial pose motion state vector is set as... The target pose motion state vector is ,in This represents the initial joint angular displacement. The initial joint angular velocity, The initial joint angular acceleration, For the target joint angular displacement, For the target joint angular velocity, The target joint angular acceleration is defined. Based on the kinematic conservation rules of the start and end states, a global boundary condition containing six independent constraints is established, and a boundary condition vector is constructed accordingly. .

[0041] Step 2: Kinematic Extremum Analysis and Adaptive Shortest Time Solution

[0042] This step is based on the aforementioned physical limit constraint parameters. , and The global running time of the trajectory is adaptively tuned to ensure that the robotic arm stays within the safe kinematic boundaries of the motor in real time when performing high-speed linkage tasks.

[0043] For typical stationary-to-stationary working conditions in single-segment point-to-point motion of industrial robots (i.e., the velocity and acceleration are both zero at the initial and final moments), the single-segment joint angular displacement span is set as follows: Based on the derivative equation of the fifth-order polynomial basis functions established in step one, and through extreme value characteristic analysis of the trajectory evolution over the entire time period, it can be seen that in the open interval... Internal, theoretical peak joint angular velocity Theoretical peak angular acceleration and the theoretical peak accelerometer They are as follows:

[0044]

[0045] in, The span of single-segment joint angular displacement The absolute value of.

[0046] To meet the system's hard physical constraints, the peak values ​​of each kinematic parameter must be less than their corresponding maximum allowable threshold. , and Based on the above extreme value distribution patterns, the theoretical minimum running time limits allowed under kinematic extreme value constraints of each order are derived:

[0047]

[0048] in, This represents the time limit corresponding to the speed limit constraint. This represents the time limit corresponding to the acceleration limit constraint. This is the time limit corresponding to the jerk limit constraint.

[0049] The above calculation process establishes a nonlinear mapping relationship between running time and the physical limits of the motor, ensuring the physical feasibility of the planning results. To simultaneously satisfy physical constraints of all orders and select the most stringent constraints, the system's adaptive global theoretical minimum running time is determined. Take the maximum value of the above three time limits:

[0050]

[0051] in, To achieve the adaptive global theoretical minimum running time, in specific engineering deployments, to adapt to the digital discretization control requirements of the underlying motion controller, the theoretical minimum time obtained through the above nonlinear constraints will be used. Perform an up rounding operation and align it precisely to the discretized control interpolation cycle of the underlying controller. The time is an integer multiple of the time, thus obtaining the discretized runtime finally applied to online solution. :

[0052] Step 3: Analytical solution of fifth-degree polynomial coefficients and timing optimization of underlying interpolation

[0053] Substituting the global boundary conditions constructed in step one into the fifth-degree polynomial and its derivative equations, we construct a linear matrix equation for solving the polynomial coefficients:

[0054]

[0055] in The coefficient vector is formed by the coefficients of the undetermined polynomials; For only discretization runtime The relevant 6x6 Vandermonde matrix has the following specific matrix form:

[0056]

[0057] In the actual engineering control of industrial robots, due to the discrete running time The above matrix is ​​always greater than zero. It is always a non-singular matrix and necessarily has a definite analytic inverse matrix. For the robotic arm's work condition of moving from a stationary to a stationary state, substituting the single-segment joint angular displacement span... With discretization runtime The underlying planner can directly calculate the explicit analytical solution of the coefficients of the fifth-degree polynomial trajectory using a defined algebraic formula:

[0058]

[0059] The aforementioned interpolation framework possesses the characteristics of non-iterative and branch-free analytical computation. Within the discretization interpolation cycle of the underlying controller, the planner can generate trajectory points in real time using only basic hardware multiply-accumulate instructions. Compared to traditional high-order S-curve solutions that heavily rely on transcendental equation iteration and complex nonlinear piecewise logic, the explicit analytical solution architecture adopted in this step completely eliminates the timing fluctuations induced by conditional branch jumps from the underlying computational mechanism.

[0060] In the benchmark test of the control cycle execution overhead for continuously generating one million trajectory interpolation points in this embodiment, the average single-step time of the fifth-order polynomial interpolation algorithm of this invention is 0.6143 microseconds, which is 26.8% shorter than that of the traditional seven-segment S-curve; the maximum single-step observation time is compressed to 0.6852 microseconds, and the standard deviation of the time is converged to 0.0156 microseconds.

[0061] Step 4: Closed-loop execution of trajectory commands and smooth multi-axis linkage based on distributed control architecture

[0062] The discretized trajectory command sequence generated in step three is sent to the underlying control system of the industrial robot for real-time closed-loop execution. In this embodiment, the control system adopts a distributed decoupled architecture, which includes an application planning layer, a communication layer, and an execution layer from top to bottom.

[0063] The application planning layer, based on the aforementioned fifth-order polynomial interpolation strategy, generates joint spatial trajectory commands containing position, velocity, and acceleration online according to a set 125 Hz high-frequency interpolation period. These commands are sent to the underlying execution layer via an asynchronous communication mechanism. Upon receiving the commands, the execution layer's general-purpose joint trajectory controller uses a proportional-integral-derivative control law for servo tracking and provides high-frequency feedback on the joint state, forming a fully closed-loop control. This distributed architecture effectively isolates the upper-layer trajectory planning from the lower-layer servo control module, ensuring that the smoothness of the trajectory commands is transmitted to the physical execution end without loss or delay. The closed-loop control system architecture of this embodiment is as follows: Figure 2 As shown.

[0064] Under the aforementioned closed-loop control architecture, the industrial robot is controlled to perform multi-axis spatial linkage point-to-point operations. Taking the core large-inertia shoulder joint of the six-degree-of-freedom robotic arm, which bears the most significant coupling between Coriolis force and centrifugal force, as an example, its response evolution process when executing discretized trajectory commands is as follows: Within the open interval of command execution, the position and velocity of the joint maintain a smooth transition without any operational breakpoints. The acceleration exhibits a smooth cubic polynomial characteristic distribution throughout the time domain. Compared with the trapezoidal acceleration profile and slope abrupt changes that inevitably exist in traditional S-curve planning algorithms, the trajectory executed by this invention achieves a globally smooth, stepless transition of acceleration within the kinematic extreme boundary. This significantly reduces the root mean square value of jerk, a core indicator characterizing flexible impact, with a reduction of 63.9% for the shoulder joint and over 68% for the base and elbow, thereby substantially reducing the multi-body flexible impact of the robotic arm at the physical execution level. In this embodiment, the time-domain response evolution processes of the position, velocity, and acceleration of the core joint executing the trajectory commands of this invention are as follows: Figure 3 , Figure 4 , Figure 5 As shown.

[0065] Furthermore, the trajectory command of this invention exhibits significant frequency domain vibration suppression characteristics during closed-loop execution. In the mid-frequency transition and energy leakage region of command energy release from 0.5 Hz to 3.0 Hz, the execution trajectory of this invention demonstrates excellent high-frequency roll-off characteristics. Compared to the frequency domain resonance peaks and persistent oscillations caused by acceleration trapezoidal segmentation in traditional algorithms, the control command of this invention completely eliminates the aforementioned high-frequency excitation components from the input end, and the command energy rapidly decays with increasing frequency, approaching the system noise floor. The acceleration frequency domain characteristics during execution in this embodiment are as follows: Figure 6 As shown.

[0066] Furthermore, in actual industrial production, changes in the load at the end effector of the robotic arm can cause significant perturbations in the system's dynamic parameters. This embodiment, when dealing with multiple varying load conditions such as 0 kg no load, 2.5 kg half load, and 5 kg full load, significantly mitigates the transient impact of inertia variations on the underlying servo controller through the aforementioned highly smooth trajectory input. Under independent closed-loop conditions without introducing additional dynamic feedforward compensation, the system can quickly converge and enter a stable following state after passing through a very short startup transient. The peak dynamic tracking error during the entire main motion and deceleration phases is strictly limited to a very small range. Data shows that the maximum dynamic tracking error under the 5 kg full load condition is only 0.0180 rad, with no obvious trajectory divergence or high-frequency oscillation. The dynamic tracking error performance of the joint controlled by this invention under different load conditions is as follows: Figure 7 As shown.

[0067] Example 2

[0068] This embodiment is basically the same as Embodiment 1, except that the application scenario is multi-axis linkage tool trajectory planning for CNC machine tools. The fifth-order polynomial smooth trajectory generation method of this invention is deployed in the CNC system. By limiting the jerk, high-frequency oscillations in the feed system are prevented, thereby improving the surface quality of high-precision curved surface machining. The specific execution steps are the same as in Embodiment 1, and will not be repeated here.

[0069] Example 3

[0070] This embodiment is basically the same as Embodiment 1, except that the application scenario is the smooth transition of local paths and speed planning for a mobile robot. This method is applied to the acceleration and deceleration control of a wheeled robot to achieve smooth movement and prevent the load from violently shaking or tipping over due to inertial impact. The specific execution steps are the same as in Embodiment 1, and will not be repeated here.

[0071] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A method for generating high-order smooth trajectories for industrial robots, characterized in that, The method for generating high-order smooth trajectories for industrial robots includes: Step 1: Construct a trajectory planning model with a fifth-order polynomial as the basis function, establish the angular displacement function and its kinematic derivative equations of each order; set the motion state at the start and end times, and establish global boundary conditions; Step 2: Based on the physical limit constraint parameters, the global running time of the trajectory is adaptively tuned, the theoretical shortest running time that satisfies each order of physical limit constraints is calculated, and this time is aligned to an integer multiple of the discretized control interpolation period of the underlying controller to obtain the discretized running time. Step 3: Substitute the global boundary conditions into the fifth-order polynomial and its derivative equations to construct a linear matrix equation. Substitute the discretized running time into the equation and directly solve the fifth-order polynomial trajectory coefficients using algebraic formulas to generate a discretized trajectory instruction sequence. Step 4: Send the discretized trajectory command sequence to the underlying control system of the industrial robot for real-time closed-loop execution.

2. The method for generating high-order smooth trajectories for industrial robots according to claim 1, characterized in that, The motion states at the start and end times mentioned in step 1 are specifically as follows: Start time and the end time ,in For runtime variables, To discretize the runtime; Set the initial pose motion state vector as The target pose motion state vector is ,in This represents the initial joint angular displacement. The initial joint angular velocity, The initial joint angular acceleration, For the target joint angular displacement, For the target joint angular velocity, The global boundary conditions for the target joint angular acceleration consist of six independent constraints, based on which a boundary condition vector is constructed. .

3. The method for generating high-order smooth trajectories for industrial robots according to claim 1, characterized in that, The physical limit constraint parameters include the maximum permissible speed of the joint. Maximum permissible acceleration and maximum permissible jerk .

4. The method for generating high-order smooth trajectories for industrial robots according to claim 1, characterized in that, Step 2 addresses a typical stationary-to-stationary working condition in a single-segment point-to-point motion of an industrial robot, where both velocity and acceleration are zero at the initial and final moments. The single-segment joint angular displacement span is set to... ,in The span of angular displacement; In the open interval Internal, theoretical peak joint angular velocity Theoretical peak angular acceleration and the theoretical peak accelerometer They are respectively: ； The specific minimum theoretical running time limits allowed under each order of physical limit constraints are as follows: ； in, This is the time limit corresponding to the speed limit. This is the time limit corresponding to the acceleration limit. The time limit corresponding to the jerk limit. The span of single-segment joint angular displacement The absolute value; The system's adaptive global theoretical shortest running time Take the maximum value of the above three time limits: ； in, To adapt to the global theoretical shortest running time; theoretical shortest running time Round up and align to the discretized control interpolation period of the underlying controller. The integer multiple of the given time is used to obtain the discretized runtime finally applied to the online solution. .

5. The method for generating high-order smooth trajectories for industrial robots according to claim 1, characterized in that, The linear matrix equation mentioned in step three is: ,in, For the coefficients of the undetermined polynomial The coefficient vector formed For only discretization runtime The relevant 6x6 Vandermonde matrix has the following specific matrix form: ； For the robotic arm's work condition of moving from a stationary state to a stationary state, the single-segment joint angular displacement span is substituted. With discretization runtime The underlying planner calculates the explicit analytical solution of the coefficients of the fifth-degree polynomial trajectory using a defined algebraic formula: 。 6. The method for generating high-order smooth trajectories for industrial robots according to claim 1, characterized in that, The underlying control system described in step 4 is a distributed decoupled architecture, which includes, from top to bottom, the application planning layer, the communication layer, and the execution layer. Based on the aforementioned fifth-order polynomial interpolation strategy, the application planning layer generates joint spatial trajectory commands containing position, velocity, and acceleration online according to a set 125 Hz high-frequency interpolation period. The commands are sent to the execution layer via an asynchronous communication mechanism. After receiving the commands, the general joint trajectory controller of the execution layer uses a proportional-integral-derivative control law for servo tracking and provides high-frequency feedback on the joint status to form a full closed-loop control.

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