Robot online strict time-trajectory tracking method based on arc length driving

Abstract

The present application belongs to the technical field of trajectory tracking, and particularly relates to a robot online strict on-time trajectory tracking method based on arc length driving, steps of which comprise: collecting environment input information; calculating arc length increment of adjacent discrete points and arc length node sequence; in each sampling control period, adopting fixed time sliding mode control to correct expected linear velocity, controlling robot to move with the corrected expected linear velocity; obtaining linear velocity amplitude and expected heading angle; designing angular velocity control law to realize heading adjustment; estimating instantaneous curvature, applying discrete first-order change rate constraint to the limited expected linear velocity and uniformized angular velocity to obtain smooth instruction to control robot to move; when the remaining arc length is less than the arc length threshold, adopting linear interpolation to gradually lower the linear velocity to the lower limit of linear velocity, and repeatedly executing the control flow in each sampling control period until the robot reaches the target position. The present application can consider on-time arrival and path geometry invariance, and has small calculation cost.

Description

Technical Field

[0001] This invention belongs to the field of trajectory tracking technology, specifically relating to an online strict time-based trajectory tracking method for robots based on arc length drive. Background Technology

[0002] Mobile robots are widely used in logistics, inspection, and unmanned material handling. Their navigation systems typically consist of global planning, local following, and obstacle avoidance modules. In practical tasks, on-time arrival (ETA) is one of the core requirements, demanding that robots accurately schedule their time progress while ensuring geometric trajectory tracking accuracy. To achieve this goal, the industry has explored various technical directions: arc length parameterization technology can decouple path shape from time, facilitating the allocation of velocity along the path tangentially and time shaping; sliding mode control has strong robustness, among which fixed-time sliding mode control (FTSMC) is particularly suitable for time-constrained scenarios because its convergence time is independent of the initial error. In addition, engineering implementation must also meet various physical constraints such as linear velocity, angular velocity, linear acceleration, angular acceleration, and lateral acceleration, and correct for real-time drift during execution to ensure operational stability.

[0003] However, existing methods still have several shortcomings and are difficult to meet engineering requirements: First, they lack online optimization capabilities. Offline time shaping methods cannot adapt to disturbance and temporary obstacle avoidance scenarios, while online optimization schemes face problems such as high computational overhead and sensitivity to initial values, easily leading to infeasible solutions. Second, geometric tracking and time scheduling are disconnected. Conventional geometric tracking lacks closed-loop control for arrival time limits, and cannot catch up with the preset time progress in time after encountering disturbances. Third, the parameterization method is unreasonable. When arc length is not used as the progress variable, uneven sampling will cause the speed to fluctuate, and the lateral amplitude limit is difficult to maintain consistency, which in turn leads to path deformation and operation comfort issues. Fourth, constraint processing and curvature control are imperfect. There is a lack of clear constraint processing order and curvature preservation principles, which easily causes control quantity jitter. Fifth, the time accumulation method has deviations. Using simulated step size instead of real time accumulation easily produces beat drift, directly undermining the robot's ability to arrive on time. Summary of the Invention

[0004] In view of the shortcomings of the prior art, the purpose of this invention is to provide a strict time-arrival trajectory tracking method for robots based on arc length drive, which can take into account both on-time arrival and path geometry invariance, and the core operations are interpolation, algebra and amplitude limiting, with low computational overhead and can run in real time on embedded platforms.

[0005] To achieve the above objectives, this invention provides a method for online strict time-to-time trajectory tracking of robots based on arc length drive, comprising the following steps:

[0006] S1. Collect environmental input information, including the robot's current pose information, velocity status, reference trajectory set, constraint parameters, sampling control cycle and preset arrival time;

[0007] S2. Calculate the arc length increment and arc length node sequence of adjacent discrete points in the reference trajectory set, and perform position interpolation to obtain continuous position function and continuous tangential function;

[0008] S3. In each sampling control cycle, calculate the actual progress of the robot's movement, calculate the expected linear velocity based on the remaining arc length and remaining time, construct the expected arc length linearly according to the actual progress based on the continuous position function, compare the actual progress with the expected arc length, use fixed-time sliding mode control to correct the expected linear velocity, and use the corrected expected linear velocity to control the robot's movement.

[0009] S4. Using a continuous tangential function, the desired linear velocity is projected onto the reference trajectory in the unit tangential direction at the current arc length position, and a virtual velocity component is formed with error feedback to obtain the linear velocity amplitude and the desired heading angle.

[0010] S5. Track the difference between the robot's actual heading angle and the desired heading angle, construct the minimum angle difference, generate a reference angular velocity feedforward, and then design an angular velocity control law to achieve heading adjustment;

[0011] S6. Estimate the instantaneous curvature based on the linear velocity amplitude and angular velocity command, perform time scaling and amplitude limiting on the desired linear velocity, and design and limit the curvature-based uniform angular velocity.

[0012] S7. Apply discrete first-order rate of change constraints to the desired linear velocity and uniform angular velocity after amplitude limiting to obtain smoothing commands, and use smoothing commands to control the robot's movement.

[0013] S8. Set an arc length threshold. When the remaining arc length is less than the arc length threshold, use linear interpolation to gradually reduce the linear velocity to the lower limit of the linear velocity. Repeat S3-S7 in each sampling control cycle until the robot reaches the target position.

[0014] As a preferred embodiment of the present invention, in S1, the collected environmental input information is represented as follows:

[0015] ;

[0016] In the formula, Represents a set of environmental input information; This indicates the robot's current pose information, including its current coordinates. and current heading angle ; This indicates the robot's current speed state, including its current linear velocity. and current angular velocity ; Spatial coordinates of the reference trajectory; set of reference trajectories Let N be a discrete set of points, with index i for each point. The spatial coordinates of discrete point i in the reference trajectory are: ; This represents the velocity constraint parameters, including the lower limit of linear velocity. Upper limit of linear velocity and the upper limit of angular velocity amplitude ; This represents acceleration constraint parameters, including the upper limit of linear acceleration. and upper limit of angular acceleration ; Represents time parameters, including the sampling control period. and preset arrival time .

[0017] As a preferred embodiment of the present invention, in S2, the calculation of the arc length increment of adjacent discrete points and the arc length node sequence is expressed as follows:

[0018] ;

[0019] ;

[0020] In the formula, This represents the arc length increment from discrete point i to discrete point i+1. The spatial coordinates corresponding to the discrete point i+1 in the reference trajectory are: ; The node representing the arc length is the cumulative arc length from the starting point to the discrete point i; This represents the arc length increment from discrete point j to discrete point j+1. ;

[0021] Position interpolation is performed using the PCHIP method, specifically, for any interval... The interpolation polynomial is defined as follows:

[0022] ;

[0023] ;

[0024] In the formula, S is any continuous arc length. , , For continuous position functions, y and x correspond to the reference trajectory at point S, respectively; The node represents the arc length, which is the cumulative arc length from the starting point to the discrete point i+1; the superscript T indicates transpose. This indicates the arc length span of the current segment; This represents the normalized local parameters of the current segment; , They represent , The node slope at discrete point i; , They represent , The node slope at discrete point i+1;

[0025] Based on the unnormalized tangent vector at each discrete point i, the unit tangent vector sample is calculated. , , These represent the components of the discrete point i corresponding to the current arc length position in the x and y directions, respectively, which determine the direction of the velocity. Then, PCHIP interpolation is performed on it to obtain the continuous tangential function.

[0026] As a preferred embodiment of the present invention, in S3, during the k-th sampling control cycle, the desired linear velocity is calculated based on the remaining arc length and the remaining time, expressed as:

[0027] ;

[0028] In the formula, Let be the desired linear velocity in the k-th sampling control cycle; The total arc length of the trajectory; This represents the cumulative arc length traveled by the robot during the kth sampling control cycle; This represents the remaining arc length of the k-th sampling control cycle; Let be the actual travel time of the robot during the kth sampling control cycle; , The sliding mode gain parameter has a value greater than 0; It is a saturation function; , The power exponent converges in a fixed time. .

[0029] As a preferred embodiment of the present invention, in S4, during the k-th sampling control cycle, the desired linear velocity is projected onto the reference trajectory in the unit tangential direction at the current arc length position to obtain reference velocities in both the x and y directions, and virtual velocity components are formed using error feedback:

[0030] ;

[0031] ;

[0032] ;

[0033] In the formula, , These represent the components of the desired linear velocity in the x and y directions during the k-th sampling control cycle, respectively. The planar position error during the kth sampling control cycle; , They represent The x and y components; , These represent the x and y coordinates of the robot's current position during the k-th sampling control cycle; , These represent the x and y coordinates of the robot's reference position during the k-th sampling control cycle; The virtual velocity after adding position error feedback during the kth sampling control cycle; , These represent the virtual velocity components in the x and y directions after adding position error feedback during the k-th sampling control cycle; , This is the feedback gain parameter for the position error, and its value is greater than 0.

[0034] The linear velocity amplitude at the kth sampling control cycle is obtained. With desired heading angle , is represented as:

[0035] ;

[0036] ;

[0037] In the formula, atan2 represents the arctangent function in the four quadrants.

[0038] As a preferred embodiment of the present invention, in step S5, the minimum angle difference is calculated as follows:

[0039] ;

[0040] In the formula, This represents the minimum surround angle error, i.e., the minimum angle difference, during the k-th sampling control cycle. For mapping to Angle difference calculation; The actual heading angle of the robot during the k-th sampling control cycle; It is defined as;

[0041] The calculation method for the reference angular velocity feedforward is as follows:

[0042] ;

[0043] In the formula, The reference angular velocity feedforward at the kth sampling control cycle; This represents the expected heading angle during the (k-1)th sampling control cycle;

[0044] Design an angular velocity control law, generate angular velocity commands, and achieve heading adjustment, as shown below:

[0045] ;

[0046] In the formula, This is the angular velocity command during the kth sampling control cycle; , is the sliding mode gain parameter in the angular velocity control law; p is the low-order exponent. ; q represents a high-order exponent. .

[0047] In a preferred embodiment of the present invention, the instantaneous curvature in S6 is expressed as:

[0048] ;

[0049] In the formula, The instantaneous curvature at the k-th sampling control cycle; A constant to prevent the denominator from being zero;

[0050] The desired linear velocity is time-scaled and limited, expressed as:

[0051] ;

[0052] In the formula, This represents the linear velocity after time scaling and limiting during the kth sampling control cycle; This represents the amplitude limiting control function;

[0053] The design is based on curvature-uniformed angular velocity with amplitude limiting, expressed as:

[0054] ;

[0055] In the formula, This is the uniform angular velocity during the k-th sampling control cycle.

[0056] As a preferred embodiment of the present invention, in S7, a linear acceleration, angular acceleration, and lateral stability constraint controller is constructed to... , Applying a discrete first-order rate of change constraint yields a smoothing command, expressed as:

[0057] ;

[0058] In the formula, , The smoothing commands represent the smoothed linear velocity and angular velocity commands for the k-th sampling control cycle, respectively. , These represent the linear velocity and angular velocity commands during the (k-1)th sampling control cycle, respectively.

[0059] As a preferred embodiment of the present invention, in S7, the normal acceleration of planar motion during the k-th sampling control cycle is considered. Upper limit constraint ,in Assuming an upper limit for normal acceleration, the linear velocity is uniformly clipped, expressed as:

[0060] ;

[0061] In the formula, This is the smoothed linear velocity command after clipping during the kth sampling control cycle.

[0062] As a preferred embodiment of the present invention, in S8, when the remaining arc length is less than the arc length threshold, linear interpolation is used to gradually reduce the linear velocity to the lower limit of the linear velocity, which is expressed as:

[0063] ;

[0064] In the formula, The optimized linear velocity command when the remaining arc length is less than the arc length threshold; This is the arc length threshold.

[0065] The beneficial effects of this invention are:

[0066] This invention constructs an integrated online trajectory tracking framework encompassing arc length, time, control, and constraints. PCHIP interpolation, using arc length as a unified parameter, constructs a continuous and distortion-free unit tangential field, avoiding the jumps inherent in conventional spline overshoot and point-by-point monitoring. The time layer (calculating the desired linear velocity based on the remaining arc length and remaining time, and correcting the desired linear velocity using fixed-time sliding mode control) eliminates the need for offline reparameterization or numerical optimization. Based on the nominal average velocity of the remaining arc length / remaining time and a progress error compensator inspired by FTSMC, it achieves fixed-time convergence and explicit velocity scheduling independent of initial values. Simultaneously, the curvature-preserving time scaling strategy only scales the linear velocity and generates angular velocities according to curvature consistency, balancing on-time arrival with path geometric invariance. Furthermore, the core operations are interpolation, algebra, and amplitude limiting, resulting in low computational overhead. It can run in real-time on embedded platforms, is suitable for real-world scenarios with only discrete points and no analytical trajectory, and can be directly integrated with existing global / local planners.

[0067] This invention unifies linear / angular velocity rate limiting, lateral acceleration adaptive speed limiting, and terminal linear easing into a consistent limiting sequence during control execution, ensuring that velocity, acceleration, and lateral stability are synergistically satisfied. Attitude control employs a fixed-time sliding mode, enhancing robustness to modeling errors and disturbances. Test data shows that after introducing the time layer, the average arrival time error decreased from approximately 3.36s to 0.18s (a reduction of 94.6%), the arrival rate (±0.5s determination) increased from 6.7% to 98.3%, and the average position error remained at a low level of approximately 0.11m. This achieves a significant improvement in on-time arrival capability and robustness without sacrificing spatial tracking accuracy, balancing safety, comfort, and engineering feasibility. Attached Figure Description

[0068] Figure 1 This is a flowchart illustrating the principle of this invention;

[0069] Figure 2 This is a graph showing the results of a tracking experiment in Embodiment 1 of the present invention. Detailed Implementation

[0070] The embodiments of the present invention will be further described below with reference to the accompanying drawings:

[0071] Example 1: As Figure 1 As shown, the online strict time-to-time trajectory tracking method for robots based on arc length drive includes the following steps:

[0072] S1. Collect environmental input information, including the robot's current pose information, velocity status, reference trajectory set, constraint parameters, sampling control cycle and preset arrival time;

[0073] S2. Calculate the arc length increment and arc length node sequence of adjacent discrete points in the reference trajectory set, and perform position interpolation to obtain continuous position function and continuous tangential function;

[0074] S3. In each sampling control cycle, calculate the actual progress of the robot's movement, calculate the expected linear velocity based on the remaining arc length and remaining time, construct the expected arc length linearly according to the actual progress based on the continuous position function, compare the actual progress with the expected arc length, use fixed-time sliding mode control to correct the expected linear velocity, and use the corrected expected linear velocity to control the robot's movement.

[0075] S4. Using a continuous tangential function, the desired linear velocity is projected onto the reference trajectory in the unit tangential direction at the current arc length position, and a virtual velocity component is formed with error feedback to obtain the linear velocity amplitude and the desired heading angle.

[0076] S5. Track the difference between the robot's actual heading angle and the desired heading angle, construct the minimum angle difference, generate a reference angular velocity feedforward, and then design an angular velocity control law to achieve heading adjustment (smooth and without jumps).

[0077] S6. Estimate the instantaneous curvature based on the linear velocity amplitude and angular velocity command, perform time scaling and amplitude limiting on the desired linear velocity, and design and limit the curvature-based uniform angular velocity.

[0078] S7. Apply discrete first-order rate of change constraints to the desired linear velocity and uniform angular velocity after amplitude limiting to obtain smoothing commands, and use smoothing commands to control the robot's movement.

[0079] S8. Set an arc length threshold. When the remaining arc length is less than the arc length threshold, use linear interpolation to gradually reduce the linear velocity to the lower limit of the linear velocity. Repeat S3-S7 in each sampling control cycle until the robot reaches the target position.

[0080] In S1, the collected environmental input information is represented as follows:

[0081] ;

[0082] In the formula, Represents a set of environmental input information; This indicates the robot's current pose information, including its current coordinates. and current heading angle ; This indicates the robot's current speed state, including its current linear velocity. and current angular velocity ; Spatial coordinates of the reference trajectory; set of reference trajectories Let N be a discrete set of points, with index i for each point. i=0 corresponds to the starting point, and the spatial coordinates of the discrete point i in the reference trajectory are: ; This represents the velocity constraint parameters, including the lower limit of linear velocity. Upper limit of linear velocity and the upper limit of angular velocity amplitude ; This represents acceleration constraint parameters, including the upper limit of linear acceleration. and upper limit of angular acceleration ; Represents time parameters, including the sampling control period. and preset arrival time .

[0083] In S2, the calculation of the arc length increment of adjacent discrete points and the arc length node sequence are represented as follows:

[0084] ;

[0085] ;

[0086] In the formula, This represents the arc length increment from discrete point i to discrete point i+1. The spatial coordinates corresponding to the discrete point i+1 in the reference trajectory are: ; The node representing the arc length is the cumulative arc length from the starting point to the discrete point i; This represents the arc length increment from discrete point j to discrete point j+1. ;

[0087] Position interpolation is performed using the PCHIP method, specifically, for any interval... The interpolation polynomial is defined as follows:

[0088] ;

[0089] ;

[0090] In the formula, S is any continuous arc length. , , For continuous position functions, y and x correspond to the reference trajectory at point S, respectively; The node represents the arc length, which is the cumulative arc length from the starting point to the discrete point i+1; the superscript T indicates transpose. This indicates the arc length span of the current segment; This represents the normalized local parameters of the current segment; , They represent , The node slope at discrete point i; , They represent , The node slope at discrete point i+1; when the product of the slopes of two adjacent secant segments is ≤0, the node slope is set to 0; when the product of the slopes is greater than 0, the harmonic average of the slopes of the two secant segments is taken, which is the standard slope processing method.

[0091] Based on the unnormalized tangent vector at each discrete point i The unit tangent vector samples were calculated. :

[0092] ;

[0093] In the formula, , These represent the components of the discrete point i corresponding to the current arc length position in the x and y directions, respectively, which determine the direction of the velocity; , Let X and Y represent the components of the unnormalized tangent vector at discrete point i, respectively. This is a constant used in PCHIP interpolation to prevent the denominator from being zero; its value is... ;

[0094] Then perform PCHIP interpolation on it to obtain a continuous tangential function.

[0095] In S3, during the k-th sampling control cycle, the desired linear velocity is calculated based on the remaining arc length and remaining time, and is expressed as:

[0096] ;

[0097] In the formula, Let be the desired linear velocity in the k-th sampling control cycle; The total arc length of the trajectory; This represents the cumulative arc length traveled by the robot during the kth sampling control cycle; This represents the remaining arc length of the k-th sampling control cycle; Let be the actual travel time of the robot during the kth sampling control cycle; , The sliding mode gain parameter has a value greater than 0; in this embodiment, the values ​​are 2.0 and 5.0, respectively. For saturation functions (with boundary layers) ); , The power exponent converges in a fixed time. In this embodiment, the values ​​are 0.5 and 1.5 respectively.

[0098] After constructing the desired arc length linearly according to the actual progress, it is necessary to obtain the reference position corresponding to the desired arc length. This position is the output of the continuous position function. Fixed-time sliding mode control is an existing technology, and the desired linear velocity can be corrected using existing fixed-time sliding mode control methods.

[0099] In S4, during the k-th sampling control cycle, the desired linear velocity is projected onto the reference trajectory at the current arc length position along the unit tangential direction, resulting in reference velocities in both the x and y directions. Virtual velocity components are then formed using error feedback.

[0100] ;

[0101] ;

[0102] ;

[0103] In the formula, , These represent the components of the desired linear velocity in the x and y directions during the k-th sampling control cycle, respectively. The planar position error during the kth sampling control cycle; , They represent The x and y components; , These represent the x and y coordinates of the robot's current position during the k-th sampling control cycle; , These represent the x and y coordinates of the robot's reference position during the k-th sampling control cycle; The virtual velocity after adding position error feedback during the kth sampling control cycle; , These represent the virtual velocity components in the x and y directions after adding position error feedback during the kth sampling control cycle, respectively, which are used to uniformly generate the linear velocity magnitude and the heading target; , This is the feedback gain parameter for the position error, and its value is greater than 0. In this embodiment, the value is 1.0.

[0104] The linear velocity amplitude at the kth sampling control cycle is obtained. With desired heading angle , is represented as:

[0105] ;

[0106] ;

[0107] In the formula, atan2 represents the arctangent function in the four quadrants.

[0108] In S5, the minimum angle difference is calculated as follows:

[0109] ;

[0110] In the formula, This represents the minimum surround angle error, i.e., the minimum angle difference, during the k-th sampling control cycle. For mapping to Angle difference calculation; The actual heading angle of the robot during the k-th sampling control cycle; It is defined as;

[0111] The calculation method for the reference angular velocity feedforward is as follows:

[0112] ;

[0113] In the formula, The reference angular velocity feedforward at the kth sampling control cycle; This represents the expected heading angle during the (k-1)th sampling control cycle;

[0114] Design an angular velocity control law, generate angular velocity commands, and achieve heading adjustment, as shown below:

[0115] ;

[0116] In the formula, This is the angular velocity command during the kth sampling control cycle; , is the sliding mode gain parameter in the angular velocity control law, and in this embodiment, its values ​​are 2.0 and 1.5 respectively; p is a low-power exponent. In this embodiment, the value is 0.5; q is a high-order exponent. In this embodiment, the value is 1.5.

[0117] In S6, the instantaneous curvature is expressed as:

[0118] ;

[0119] In the formula, The instantaneous curvature at the k-th sampling control cycle; To prevent the constant from having a denominator of zero, this embodiment takes the value 10. -9 ;

[0120] The desired linear velocity is time-scaled and limited, expressed as:

[0121] ;

[0122] In the formula, This represents the linear velocity after time scaling and limiting during the kth sampling control cycle; This represents the amplitude limiting control function;

[0123] The design is based on curvature-uniformed angular velocity with amplitude limiting, expressed as:

[0124] ;

[0125] In the formula, This is the uniform angular velocity during the k-th sampling control cycle.

[0126] In S7, a linear acceleration, angular acceleration, and lateral stability constraint controller is constructed to... , Applying a discrete first-order rate of change constraint yields a smoothing command, expressed as:

[0127] ;

[0128] In the formula, , The smoothing commands represent the smoothed linear velocity and angular velocity commands for the k-th sampling control cycle, respectively. , These represent the linear velocity and angular velocity commands during the (k-1)th sampling control cycle, respectively.

[0129] In S8, when the remaining arc length is less than the arc length threshold, linear interpolation is used to gradually reduce the linear velocity to the lower limit of the linear velocity, as expressed as:

[0130] ;

[0131] In the formula, The optimized linear velocity command when the remaining arc length is less than the arc length threshold; This is the arc length threshold.

[0132] Figure 2 The tracking experiment results under trajectory constraints only (no external obstacles) are presented: the horizontal and vertical coordinates are the spatial positions of x and y, respectively; the red solid line is the reference trajectory obtained by smoothing discrete points using the PCHIP method; and the blue dashed line is the closed-loop tracking trajectory of the trajectory tracking method in this embodiment. In the initial stage (approximately x < 3), the transient error caused by the initial heading alignment is quickly suppressed by fixed-time sliding mode control, and the tracking trajectory rapidly matches the reference trajectory. The reference curvature changes most significantly in the middle stage; through the combined effect of curvature-maintaining time scaling and lateral acceleration limiting, overshoot and sawtooth oscillations are avoided, resulting in a smooth and continuous path. Throughout the entire process, linear velocity, angular velocity, linear acceleration, and angular acceleration are all limited by set thresholds, and the time layer adjusts the target linear velocity in real time according to the remaining arc length and remaining time, ensuring that the overall propulsion rate is consistent with the preset arrival time limit. In the final stage, a linear slow-stop strategy is adopted, and the velocity converges to the minimum speed without abrupt changes, resulting in minimal attitude error at the endpoint. Overall, the blue trajectory visually almost overlaps with the red reference, and the key curvature segments can also maintain a stable fit, demonstrating the comprehensive advantages of the method proposed in this embodiment in terms of on-time arrival, smoothness, and robustness.

[0133] Table 1 Comparison of tracking performance with and without time layer

[0134]

[0135] Table 1 shows the comparison of tracking performance with and without a time layer. The test was divided into three rounds: A, B, and C. In each round, the preset time and reference trajectory were adjusted, and each test was performed 20 times. Table 1 shows that after introducing the time layer (calculating the expected linear velocity based on the remaining arc length and remaining time, and correcting the expected linear velocity using fixed-time sliding mode control), the arrival performance was significantly improved in the three rounds of 20 tests: the average arrival time error decreased from about 3.36s in the control group (without a time layer) to 0.18s (a reduction of 94.6%), and the arrival rate (judged within ±0.5s) increased from 6.7% to 98.3%; at the same time, the average position error remained at the same level of about 0.11m. This indicates that the method of this embodiment, after introducing the time layer, significantly improves the on-time arrival capability and robustness without sacrificing spatial tracking accuracy, and exhibits stronger real-time performance and engineering usability under different trajectory and time limit scenarios.

[0136] Example 2: Based on Example 1, consider the normal acceleration of planar motion during the k-th sampling control cycle. Upper limit constraint ,in Assuming an upper limit for normal acceleration, the linear velocity is uniformly clipped, expressed as:

[0137] ;

[0138] In the formula, This is the smoothed linear velocity command after clipping during the k-th sampling control cycle. replace It is used for robot control, ensuring lateral stability, preventing slippage / loss of control, and avoiding sudden speed changes.

[0139] Example 3: An online, strictly time-based trajectory tracking device for robots based on arc length drive, comprising:

[0140] One or more processors;

[0141] Memory, used to store one or more computer programs;

[0142] When one or more programs are executed by one or more processors, the one or more processors perform the method in Embodiment 1 or Embodiment 2.

[0143] Example 4: A computer-readable storage medium having executable instructions stored thereon, which, when executed by a processor, cause the processor to perform the method in Example 1 or Example 2.

Claims

1. A method for online strict time-to-time trajectory tracking of robots based on arc length drive, characterized in that... Includes the following steps: S1. Collect environmental input information, including the robot's current pose information, velocity status, reference trajectory set, constraint parameters, sampling control cycle and preset arrival time; S2. Calculate the arc length increment and arc length node sequence of adjacent discrete points in the reference trajectory set, and use the PCHIP method to perform position interpolation to obtain continuous position function and continuous tangential function; S3. In each sampling control cycle, calculate the actual progress of the robot's movement, calculate the expected linear velocity based on the remaining arc length and remaining time, construct the expected arc length linearly according to the actual progress based on the continuous position function, compare the actual progress with the expected arc length, use fixed-time sliding mode control to correct the expected linear velocity, and use the corrected expected linear velocity to control the robot's movement. S4. Using a continuous tangential function, the desired linear velocity is projected onto the reference trajectory in the unit tangential direction at the current arc length position, and a virtual velocity component is formed with error feedback to obtain the linear velocity amplitude and the desired heading angle. The process is as follows: In the kth sampling control cycle, the desired linear velocity is projected onto the reference trajectory at the current arc length position along the unit tangential direction, resulting in reference velocities in both the x and y directions. Virtual velocity components are then formed using error feedback. ； ； ； In the formula, , These represent the components of the desired linear velocity in the x and y directions during the k-th sampling control cycle, respectively. The planar position error during the kth sampling control cycle; , They represent The x and y components; , These represent the x and y coordinates of the robot's current position during the k-th sampling control cycle; , These represent the x and y coordinates of the robot's reference position during the k-th sampling control cycle; The virtual velocity after adding position error feedback during the kth sampling control cycle; , These represent the virtual velocity components in the x and y directions after adding position error feedback during the k-th sampling control cycle; , This is the feedback gain parameter for the position error, and its value is greater than 0. , These represent the components of the discrete point i corresponding to the current arc length position in the x and y directions, respectively, which determine the direction of the velocity; Let be the desired linear velocity in the k-th sampling control cycle; The linear velocity amplitude at the kth sampling control cycle is obtained. With desired heading angle , is represented as: ； ； In the formula, atan2 represents the arctangent function in the four quadrants; S5. Track the difference between the robot's actual heading angle and the desired heading angle, construct the minimum angle difference, generate a reference angular velocity feedforward, and then design an angular velocity control law to achieve heading adjustment; S6. Estimate the instantaneous curvature based on the linear velocity amplitude and angular velocity command, perform time scaling and amplitude limiting on the desired linear velocity, and design and limit the curvature-based uniform angular velocity. S7. Apply discrete first-order rate of change constraints to the desired linear velocity and uniform angular velocity after amplitude limiting to obtain smoothing commands, and use smoothing commands to control the robot's movement. S8. Set an arc length threshold. When the remaining arc length is less than the arc length threshold, use linear interpolation to gradually reduce the linear velocity to the lower limit of the linear velocity. Repeat S3-S7 in each sampling control cycle until the robot reaches the target position.

2. The online strictly time-based trajectory tracking method for robots based on arc length drive according to claim 1, characterized in that, In S1, the collected environmental input information is represented as follows: ； In the formula, Represents a set of environmental input information; This indicates the robot's current pose information, including its current coordinates. and current heading angle ; This indicates the robot's current speed state, including its current linear velocity. and current angular velocity ; Spatial coordinates of the reference trajectory; set of reference trajectories Let N be a discrete set of points, with index i for each point. The spatial coordinates of discrete point i in the reference trajectory are: ; This represents the velocity constraint parameters, including the lower limit of linear velocity. Upper limit of linear velocity and the upper limit of angular velocity amplitude ; This represents acceleration constraint parameters, including the upper limit of linear acceleration. and upper limit of angular acceleration ; Represents time parameters, including the sampling control period. and preset arrival time .

3. The online strictly time-based trajectory tracking method for robots based on arc length drive according to claim 2, characterized in that, In S2, the calculation of the arc length increment of adjacent discrete points and the arc length node sequence are expressed as follows: ； ； In the formula, This represents the arc length increment from discrete point i to discrete point i+1. The spatial coordinates corresponding to the discrete point i+1 in the reference trajectory are: ; The node representing the arc length is the cumulative arc length from the starting point to the discrete point i; This represents the arc length increment from discrete point j to discrete point j+1. ; Position interpolation is performed using the PCHIP method, specifically, for any interval... The interpolation polynomial is defined as follows: ； ； In the formula, S is any continuous arc length. , , For continuous position functions, y and x correspond to the reference trajectory at point S, respectively; The node represents the arc length, which is the cumulative arc length from the starting point to the discrete point i+1; the superscript T indicates transpose. This indicates the arc length span of the current segment; This represents the normalized local parameters of the current segment; , They represent , The node slope at discrete point i; , They represent , The node slope at discrete point i+1; Based on the unnormalized tangent vector at each discrete point i, the unit tangent vector sample is calculated. Then, perform PCHIP interpolation on it to obtain a continuous tangential function.

4. The online strictly time-based trajectory tracking method for robots based on arc length drive according to claim 3, characterized in that, In S3, during the k-th sampling control cycle, the desired linear velocity is calculated based on the remaining arc length and remaining time, expressed as: ； In the formula, The total arc length of the trajectory; This represents the cumulative arc length traveled by the robot during the kth sampling control cycle; This represents the remaining arc length of the k-th sampling control cycle; Let be the actual travel time of the robot during the kth sampling control cycle; , The sliding mode gain parameter has a value greater than 0; It is a saturation function; , The power exponent converges in a fixed time. .

5. The online strictly time-bound trajectory tracking method for robots based on arc length drive according to claim 4, characterized in that, In S5, the minimum angle difference is calculated as follows: ； In the formula, This represents the minimum surround angle error, i.e., the minimum angle difference, during the k-th sampling control cycle. For mapping to Angle difference calculation; The actual heading angle of the robot during the k-th sampling control cycle; It is defined as; The calculation method for the reference angular velocity feedforward is as follows: ； In the formula, The reference angular velocity feedforward at the kth sampling control cycle; This represents the expected heading angle during the (k-1)th sampling control cycle; Design an angular velocity control law, generate angular velocity commands, and achieve heading adjustment, as shown below: ； In the formula, This is the angular velocity command during the kth sampling control cycle; , is the sliding mode gain parameter in the angular velocity control law; p is the low-order exponent. ; q represents a high-order exponent. .

6. The online strictly time-bound trajectory tracking method for robots based on arc length drive according to claim 5, characterized in that, In S6, the instantaneous curvature is expressed as: ； In the formula, The instantaneous curvature at the k-th sampling control cycle; A constant to prevent the denominator from being zero; The desired linear velocity is time-scaled and limited, expressed as: ； In the formula, This represents the linear velocity after time scaling and limiting during the kth sampling control cycle; This represents the amplitude limiting control function; The design is based on curvature-uniformed angular velocity with amplitude limiting, expressed as: ； In the formula, This is the uniform angular velocity during the k-th sampling control cycle.

7. The online strictly time-based trajectory tracking method for robots based on arc length drive according to claim 6, characterized in that, In S7, a linear acceleration, angular acceleration, and lateral stability constraint controller are constructed to... , Applying a discrete first-order rate of change constraint yields a smoothing command, expressed as: ； In the formula, , The smoothing commands represent the smoothed linear velocity and angular velocity commands for the k-th sampling control cycle, respectively. , These represent the linear velocity and angular velocity commands during the (k-1)th sampling control cycle, respectively.

8. The online strictly time-bound trajectory tracking method for robots based on arc length drive according to claim 7, characterized in that, In S7, the normal acceleration of planar motion during the k-th sampling control cycle is considered. Upper limit constraint ,in Assuming an upper limit for normal acceleration, the linear velocity is uniformly clipped, expressed as: ； In the formula, This is the smoothed linear velocity command after clipping during the kth sampling control cycle.

9. The online strictly time-based trajectory tracking method for robots based on arc length drive according to claim 7, characterized in that, In S8, when the remaining arc length is less than the arc length threshold, linear interpolation is used to gradually reduce the linear velocity to the lower limit of the linear velocity, as expressed as: ； In the formula, The optimized linear velocity command when the remaining arc length is less than the arc length threshold; This is the arc length threshold.

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