Predefined time sliding mode control method for a Mecanum wheeled robot
By using a predefined time sliding mode control method, the problems of uncertain convergence time and excessive control input in Mecanum wheeled robots under system uncertainty are solved, achieving faster convergence speed and smaller control input, thereby improving the stability and efficiency of the system.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- HANGZHOU DIANZI UNIV
- Filing Date
- 2023-07-10
- Publication Date
- 2026-08-04
AI Technical Summary
When faced with system uncertainties, Mecanum wheeled robots suffer from the problem of actuator saturation due to the uncertainty of the upper limit of convergence time and the excessive control input caused by traditional sliding mode control methods.
A predefined time sliding mode control method is adopted. By accurately modeling the Mecanum wheeled robot, a predefined time sliding surface and control law are designed, and parameters are reasonably designed to achieve faster convergence speed and smaller control input.
The convergence time upper limit of the Mecanum wheeled robot system can be predefined, which avoids actuator saturation, improves the convergence speed and reduces control input.
Smart Images
Figure CN116931425B_ABST
Abstract
Description
Technical Field
[0001] This invention belongs to the field of robot trajectory tracking and relates to a predefined time sliding mode control method for a Mecanum wheeled robot. Background Technology
[0002] With the continuous development of robotics technology, mobile robots have been widely used in industrial applications, warehousing and logistics, catering and hotels, geological exploration, and other fields. Mecanum wheeled robots are an important branch of mobile robots. They consist of four special mechanical wheels arranged in a specific manner. Mecanum wheeled robots can achieve zero-radius turning and movement in any direction, thus attracting widespread attention.
[0003] The Mecanum wheeled robot system is a typical multi-input multi-output nonlinear system, and is also affected by system uncertainties such as parameter variations, nonlinear friction, unknown disturbances, mechanical coupling, and measurement noise. This makes the stable motion and fast dynamic response of the Mecanum wheeled robot a challenge.
[0004] Sliding mode variable structure control is considered one of the most effective methods for handling nonlinear system problems. However, the system settling time of the traditional terminal sliding mode method is affected by the initial value of the system state and does not have a definite upper limit to the convergence time. The fixed-time sliding mode method overcomes the problem of terminal sliding mode, but its upper limit function of convergence time is not directly related to the control parameters, making it challenging to design parameters to meet the expected convergence time range. The predefined time sliding mode method, as a branch of fixed-time sliding mode, makes the upper limit of convergence time appear as a specific parameter among the control parameters, which greatly facilitates the design of the overall system. However, it still suffers from the phenomenon that when the initial value of the system is far from the equilibrium point, the control input is too large, leading to actuator saturation.
[0005] Therefore, it is necessary to conduct research to address the aforementioned deficiencies in the existing technology and provide a solution to overcome these deficiencies. Summary of the Invention
[0006] To address the aforementioned problems, the technical solution of this invention involves first accurately modeling the Mecanum wheeled robot; then, constructing a predefined time sliding surface and a predefined time control law based on a predefined time-stabilized system to ensure the predefined time nature of the system's convergence time. Through reasonable parameter design, the proposed predefined time controller enables the Mecanum wheeled robot system to achieve faster convergence speed with smaller control inputs.
[0007] The specific technical solution is as follows: the Mecanum wheeled robot includes a main controller, radar, a six-axis accelerometer, an encoder, a motor drive circuit, and a DC motor.
[0008] The main controller controls the input magnitude and converts it into the required voltage magnitude, controls the motor drive circuit in the form of pulse width modulation waveform, and interacts with the host computer via Bluetooth.
[0009] The radar and six-axis accelerometer collect the pose information of the Mecanum wheeled robot, including X-axis and Y-axis position information and yaw angle information.
[0010] The encoder collects angular velocity information of the DC motor;
[0011] The motor drive circuit converts the pulse width modulation waveform signal into a voltage signal to control the speed of the DC motor.
[0012] The predefined time sliding mode control method for the Mecanum wheeled robot described above includes the following steps:
[0013] S10, radar, six-axis accelerometer and encoder collect motion parameters of Mecanum wheeled robot;
[0014] S20, the main controller calculates the error between the desired pose and the actual pose;
[0015] S30, Update the value of the sliding mode variable;
[0016] S40, Update sliding mode control input.
[0017] Preferably, the model equation for the Mecanum wheeled robot is as follows:
[0018]
[0019] Where ζ = [x, y, χ] T It is the robot's actual pose. Let ω be its second derivative, r be the radius of the Mecanum wheel, j0 be the nominal equivalent moment of inertia of the wheel, and ω = [ω1, ω2, ω3, ω4]. T The angular velocities of the four motors are... Let ψ be the first derivative of the robot's yaw angle ψ, b0 be the nominal viscous friction force, and d = [d1, d2, d3, d4]. T It represents the uncertainty of the lumped system, where v is the input voltage of the four corresponding motors, and h(ψ) is the input voltage of the lumped system. m H and H are the correlation matrices, specifically...
[0020]
[0021]
[0022] in, ψ is the robot's yaw angle; a represents the yaw angle from the local coordinate axis X. lBelow, represents the projected length from the geometric center of the wheel to the geometric center of the robot; b represents the distance from the wheel's geometric center to the robot's geometric center in the local coordinate axis Y. l Below, the projected length from the geometric center of the wheel to the geometric center of the robot.
[0023] Preferably, the control input u is designed to simplify the model equations.
[0024]
[0025] The model equation can then be written as
[0026]
[0027] Preferably, the motion parameters in S10 include X-axis and Y-axis position information, yaw angle information, and DC motor angular velocity information.
[0028] Preferably, the sliding surface in S30 is designed as follows:
[0029]
[0030] Where s1, s2, and s3 represent the sliding mode variables corresponding to the X direction, Y direction, and yaw angle, respectively, and 0 < ρ1 < 0.5. T c1 >0 is the predefined time for the sliding surface to converge to 0; Let denote the first derivative of the error matrix, where the error matrix e represents the robot's pose error, specifically...
[0031] e = [e1, e2, e3] T =ζ-ζ d =[xx d yy d , ψ-ψ d ] T
[0032] Where ζ = [x, y, ψ] T It is the robot's actual pose, ζ d =[x d y d , ψ d ] T sig(m) represents the robot's desired pose. k , The concept is an abbreviation of the following expression.
[0033]
[0034] Preferably, the sliding mode control input in S40 is designed as follows:
[0035]
[0036] Where u = [u1, u2, u3, u4] T This corresponds to the control input for the four wheels, where 0 < ρ2 < 1. T c2 >0 is the predefined time for the pose error to converge to 0 after the sliding mode variables converge; Let h(ψ) be the second derivative of the desired pose. m ) + M, N, and B are the correlation matrices, specifically:
[0037]
[0038]
[0039]
[0040]
[0041] Where 'a' represents the distance from the local coordinate axis X. l Below, represents the projected length from the geometric center of the wheel to the geometric center of the robot; b represents the distance from the wheel's geometric center to the robot's geometric center in the local coordinate axis Y. l Below, the projected length from the geometric center of the wheel to the geometric center of the robot; Let be the known upper bound of the lumped disturbance corresponding to any wheel; diag{x1, x2, x3} is a diagonal matrix with x1, x2, x3 as the main diagonal elements, and e1, e2, e3 are the three pose errors mentioned above.
[0042] Preferably, when α takes the value of β takes the value of At this time, the overall control input will be near the minimum point; when the initial state is far from the equilibrium point, α and β corresponding to the minimum point are greater than 1; where e i (0), i = 1, 2, 3 are the absolute values of the X-axis, Y-axis, and yaw angle pose errors at time 0; s i (0), i = 1, 2, 3 are the magnitudes of the sliding mode variables at time 0.
[0043] Preferably, a Lyapunov function is designed. Differentiation yields
[0044]
[0045] in
[0046]
[0047] The convergence time of the sliding mode variable is
[0048]
[0049] When β > 1, the convergence time T f It will be faster.
[0050] Preferably, after the sliding mode variable converges to 0, a Lyapunov function is designed. Differentiation yields
[0051]
[0052] The pose error convergence time is
[0053]
[0054] Preferably, when α>1, the convergence time T s It will be faster.
[0055] This invention offers at least the following advantages: In the trajectory tracking control of Mecanum wheeled robots, the proposed predefined time sliding mode control method allows for a freely defined upper limit of the system convergence time, greatly facilitating user design. Compared to existing predefined time sliding mode methods, the proposed method for Mecanum wheeled robots achieves faster convergence with smaller control inputs, while simultaneously improving convergence speed and avoiding actuator saturation. Attached Figure Description
[0056] Figure 1 This is a top view of a robot model of a predefined time sliding mode method for a Mecanum wheeled robot according to an embodiment of the present invention.
[0057] Figure 2 This is a flowchart illustrating the steps of a predefined time sliding mode method for a Mecanum wheeled robot according to an embodiment of the present invention.
[0058] Figure 3 This is a trajectory tracking effect diagram of a predefined time sliding mode method for a Mecanum wheeled robot according to an embodiment of the present invention;
[0059] Figure 4 This is a sliding surface diagram of a predefined time sliding mode method for a Mecanum wheeled robot according to an embodiment of the present invention;
[0060] Figure 5 This is a pose error diagram of a predefined time sliding mode method for a Mecanum wheeled robot according to an embodiment of the present invention.
[0061] Figure 6 This is a control input diagram for a predefined time sliding mode method for a Mecanum wheeled robot according to an embodiment of the present invention. Detailed Implementation
[0062] 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.
[0063] Conversely, this invention encompasses any substitutions, modifications, equivalent methods, and solutions made within the spirit and scope of the invention as defined in the claims. Furthermore, to provide a better understanding of the invention, certain specific details are described in detail below. However, those skilled in the art will fully understand the invention even without these detailed descriptions.
[0064] Figure 1 The image shows a top view of a Mecanum wheel mobile robot model. Each Mecanum wheel is driven by an independent DC motor. By adjusting the different speeds of the four motors, omnidirectional movement, such as lateral and longitudinal movement, tilting movement, and zero-turn rotation, can be achieved. The relevant coordinate system of the Mecanum wheels is as follows: Figure 1 As shown, O g X g Y g O represents the global coordinate system. l X l Y l This represents a local coordinate system with its origin O. l Let ζ be the robot's geometric center, with its Y-axis always pointing towards the vehicle's longitudinal axis. The X-axis is perpendicular to the Y-axis, forming a right-handed coordinate system. The robot's right front wheel is designated as wheel number 1, and the order is counter-clockwise. The robot's pose in the global coordinate system is denoted by ζ = [x, y, ψ]. T The pose in the local coordinate system is represented by ζ. l =[x l y l , ψ l ] T express.
[0065] Control methods include:
[0066] S10: Write a program in the Mecanum wheeled robot's main controller to control the motors, where the control input is...
[0067]
[0068] S20: The position and orientation of the Mecanum wheeled robot are collected through the various sensors mentioned above, mainly the position signal collected by radar and the yaw angle signal collected by gyroscope. The signals are then transmitted to the control chip to update the controller output.
[0069] S30: After the predefined time sliding mode controller obtains the input parameters, it calculates the control input required to make the mobile robot move along the desired trajectory according to the control equation set in S1.
[0070]
[0071] S40: Converts the control input into the desired voltage for the four Mecanum wheels, outputs the corresponding PWM wave signal, controls the motor drive circuit, drives the motor to rotate, and, with the coordinated action of the four Mecanum wheels, makes the mobile robot move along the desired trajectory. By continuously obtaining output through feedback pose signals, the Mecanum wheeled robot completes the trajectory tracking task.
[0072] The kinematic model of the Mecanum wheeled mobile robot is as follows:
[0073]
[0074] Where r represents the radius of the Mecanum wheel, and ω = [ω1, ω2, ω3, ω4] T Let a and b represent the angular velocities during the motion of the four Mecanum wheels. Figure 1 The projection length in the local coordinate system. The rotation matrix from the local coordinate system to the global coordinate system is as follows:
[0075]
[0076] The transformation relationship between the two coordinate systems is as follows:
[0077]
[0078] Combining (1) and (3), we can obtain:
[0079]
[0080] In the formula
[0081]
[0082] Where ψ m =ψ + π / 4.
[0083] Differentiating (4) yields
[0084]
[0085] in for:
[0086]
[0087] The dynamic model of the Mecanum wheeled robot is
[0088]
[0089] Where ζ = [x, y, ψ] T It is the robot's actual pose. Let ω be its second derivative, r be the radius of the Mecanum wheel, j0 be the nominal equivalent moment of inertia of the wheel, and ω = [ω1, ω2, ω3, ω4]. T The angular velocities of the four motors are... Let ψ be the first derivative of the robot's yaw angle ψ, b0 be the nominal viscous friction force, and d = [d1, d2, d3, d4]. T It represents the uncertainty of the lumped system, where v is the input voltage of the four corresponding motors, and h(ψ) is the input voltage of the lumped system. m H and H are the correlation matrices.
[0090] The system equations can then be written as:
[0091]
[0092] Where H is:
[0093]
[0094] To simplify the equations, the control input is designed as follows:
[0095]
[0096] The final system equation is:
[0097]
[0098] The design process for the control input u is as follows:
[0099] First, define the tracking error matrix as follows:
[0100]
[0101] Then the sliding surface is designed as follows:
[0102]
[0103] Where s1, s2, and s3 represent the sliding mode variables corresponding to the X direction, Y direction, and yaw angle, respectively, and 0 < ρ1 < 0.5. T c1 >0 represents the predefined time for the sliding surface to converge to 0. Let denote the first derivative of the error matrix, where the error matrix e represents the robot's pose error.
[0104] Design the switching control input (the control input component where the sliding mode variable converges to 0).
[0105]
[0106] Design equivalent control inputs (control input components that maintain the system state on the sliding surface).
[0107]
[0108] The total control input is then
[0109] u=u0+u1 (17)
[0110] Where 0 < ρ2 < 1, T c2 >0 is a predefined time for the pose error to converge to 0 after the sliding mode variables converge. Let h(ψ) be the second derivative of the desired pose. m ) + M, N, and B are the correlation matrices, and a represents the correlation from the local coordinate axis X. l Below, represents the projected length from the geometric center of the wheel to the geometric center of the robot; b represents the distance from the wheel's geometric center to the robot's geometric center in the local coordinate axis Y. l Below, the projected length from the geometric center of the wheel to the geometric center of the robot; Let be the known upper bound of the lumped disturbance corresponding to any wheel; diag{x1, x2, x3} is a diagonal matrix with x1, x2, x3 as the main diagonal elements, and e1, e2, e3 are the three pose errors mentioned above, specifically:
[0111]
[0112]
[0113]
[0114]
[0115] Based on (15) and (16), in order to minimize the control input, select
[0116]
[0117]
[0118] Where e i (0), i = 1, 2, 3 are the absolute values of the X-axis, Y-axis, and yaw angle pose errors at time 0. i (0), i = 1, 2, 3 are the magnitudes of the sliding mode variables at time 0.
[0119] To prove the convergence time of the designed controller, two Lyapunov equations are designed.
[0120]
[0121]
[0122] Taking the derivative and substituting the control input and sliding surface, we can finally obtain:
[0123]
[0124]
[0125] Then V1 and V2 will converge to 0 within a specified time, i.e., the sliding mode variable T. c1 The pose error converges to 0 before reaching T. c1 +T c2 The convergence to 0(T) c1 With T c2 This is the upper bound of the convergence time, not the actual convergence time. That is, when the initial state approaches infinity, the convergence time is T. c1 With T c1 +T c2 The actual convergence time is affected by the initial value, α, β, ρ1, ρ2. When ρ1 and ρ2 are fixed, α and β are greater than 1, and the convergence time is faster than the traditional predefined time sliding mode. When the initial value of the system is large, the optimal solution designed by (22) and (23) is often greater than 1. At this time, the controller can converge faster and has a smaller control input.
[0126] See the system workflow diagram. Figure 2 After system initialization, the values of the sliding mode variables are updated based on the error between the desired and actual poses. The corresponding adaptive rate is then selected based on these values, and the controller output is calculated. See simulation results below. Figures 3 to 6 , Figure 3 The simulation trajectory tracking diagram shows that, compared to the control group, the convergence speed of this method does not change much as the error gradually converges, while the convergence speed of the control group decreases significantly as the error converges. Figure 4 Figure 5 For the specific sliding mode variables and pose errors corresponding to the X-axis, Y-axis, and yaw angle, it can be intuitively seen that the convergence time of the proposed design scheme is less than that of the control scheme. Figure 6 The control input values for the two schemes are shown, and it can be seen that the maximum control input for the four motors in the design scheme is smaller than that in the control scheme. In summary, the controller of this invention performs better, with faster convergence speed and smaller control input.
[0127] 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 predefined time sliding mode control of a Mecanum wheeled robot, characterized by, Mecanum wheeled robots include a main controller, radar, a six-axis accelerometer, an encoder, a motor drive circuit, and a DC motor. The main controller controls the input magnitude and converts it into the required voltage magnitude, controls the motor drive circuit in the form of pulse width modulation waveform, and interacts with the host computer via Bluetooth. The radar and six-axis accelerometer collect the pose information of the Mecanum wheeled robot, including X-axis and Y-axis position information and yaw angle information. The encoder collects angular velocity information of the DC motor; The motor drive circuit converts the pulse width modulation waveform signal into a voltage signal to control the speed of the DC motor. The predefined time sliding mode control method for the Mecanum wheeled robot described above includes the following steps: S10, radar, six-axis accelerometer and encoder collect motion parameters of Mecanum wheeled robot; S20, the main controller calculates the error between the desired pose and the actual pose; S30, Update the value of the sliding mode variable; S40, Update sliding mode control input; The model equations for the Mecanum wheeled robot are as follows: ； in, It is the robot's actual pose. Its second derivative, It is the radius of the Mecanum wheel. Let be the nominal equivalent moment of inertia of the wheel. The angular velocities of the four motors are... For the robot's yaw angle The first derivative, This is the nominal viscous friction force. It is the uncertainty of the lumped system. These are the input voltages of the four corresponding motors. and The correlation matrix is as follows: ； ； wherein, , is a yaw angle of the robot; denotes the projection length from the geometric center of the wheel in the local coordinate axis down to the geometric center of the robot; denotes the projection length from the geometric center of the wheel in the local coordinate axis down to the geometric center of the robot; Design control input To simplify model equations ； The model equations are then written as ； The motion parameters in S10 include X-axis and Y-axis position information, yaw angle information, and DC motor angular velocity information; The sliding surface in S30 is designed as follows: ； where, respectively represent the sliding mode variables corresponding to the X direction, Y direction and yaw angle, , , , is the predefined time for the sliding surface to converge to 0; denotes the first order derivative of the error matrix, the error matrix denotes the pose error of the robot, specifically ； wherein, is the actual pose of the robot, is the desired pose of the robot; The concept of is a shortening of the following expression ； The sliding mode control input in S40 is designed as follows: ； in, It corresponds to the control input for the four wheels. , , , This is a predefined time for the pose error to converge to 0 after the sliding mode variables converge; Let the second derivative be the desired pose. , , , The correlation matrix is as follows: ； ； ； ； in, Indicates from the local coordinate axis Below, the projected length from the geometric center of the wheel to the geometric center of the robot; Indicates from the local coordinate axis Below, the projected length from the geometric center of the wheel to the geometric center of the robot; Let be the known upper bound of the lumped disturbance corresponding to any wheel; Therefore , , A diagonal matrix with elements on the main diagonal. , , These are the three pose errors mentioned above.
2. The method of claim 1, wherein, when Values , Values When the initial state is far from the equilibrium point, the overall control input will be near the minimum point; when the initial state is far from the equilibrium point, the minimum point corresponds to... and Greater than 1; where The absolute values of the X-axis, Y-axis, and yaw angle pose errors at time 0; The value is the magnitude of the sliding mode variable at time 0.
3. The method of claim 2, wherein, designing a lyapunov function the derivative of which gives ； in, ； The convergence time of the sliding mode variable is ； When , the convergence time will be faster.
4. The method of claim 3, wherein, When the sliding mode variable converges to 0, a Lyapunov function is designed , and the derivative can be obtained ； The pose error convergence time is 。 5. The method of claim 4, wherein, when Convergence time It will be faster.