A four-legged robot model predictive control method fusing extended state observer

Abstract

The present application belongs to the technical field of intelligent robot control, and particularly relates to a four-legged robot model predictive control method fusing an extended state observer, which comprises the following steps: firstly, a dynamic model of the four-legged robot system is established, and a position ESO is designed to estimate disturbances in the translational direction in real time, and an attitude ESO is designed to estimate disturbances in the rotational direction in real time; secondly, disturbance estimations output by the ESO are used to construct a discrete state space model containing a disturbance compensation term, and on the basis, a MPC optimization problem with constraints is designed, including consideration of friction cone constraints, swing leg force constraints, upper and lower joint torque constraints and the like; finally, the MPC optimization problem is solved on line to output ground reaction forces and joint torques, so as to realize trajectory tracking control of the four-legged robot under unknown disturbance conditions. The present application can effectively suppress system uncertainties caused by unmodeled loads, external disturbances or uneven ground, and improve stability and adaptability of the robot in complex environments.

Description

Technical Field

[0001] This invention belongs to the field of intelligent robot control technology, specifically relating to a model predictive control method for a quadruped robot that integrates an extended state observer. Background Technology

[0002] Quadruped robots play a vital role in inspection, transportation, and rescue tasks in complex environments due to their superior terrain adaptability and dynamic stability. In recent years, numerous advanced quadruped robot systems, such as ANYmal, MiniCheetah, Boston Dynamics Spot, and Unitree Go1, have emerged, driving the development of legged robot technology. However, in practical applications, uncertainties such as complex terrain, load variations, and external disturbances place higher demands on robot control systems. Achieving robust, precise, and real-time motion control has become a key issue in quadruped robot research.

[0003] To address these challenges, researchers have developed various control methods. Early methods, such as control strategies based on zero-torque point (ZMP) and spring-loaded inverted pendulum (SLIP) models, achieve basic gait stability by simplifying the dynamic model, but struggle to cope with rapidly changing dynamic environments. Virtual model control (VMC) utilizes virtual springs and damping elements to construct control forces, enabling flexible adjustment of robot posture and trajectory, and possesses some terrain adaptability. Whole-body control (WBC) optimizes the overall joint torque distribution in the task space, achieving coordinated control in highly redundant systems, suitable for multi-task parallel execution. Reinforcement learning (RL) methods, with their policy self-learning capabilities, enable robots to possess a certain degree of environmental perception and motion transfer ability, but suffer from practical limitations such as large training data volume, low sample efficiency, and weak interpretability, making it difficult to widely apply to task scenarios with high safety requirements.

[0004] Among the methods described above, Model Predictive Control (MPC) has become the mainstream framework for dynamic control of quadruped robots in recent years due to its ability to solve optimal control sequences online, its compatibility with system constraints, and its state prediction capabilities. MPC constructs a state-space model and addresses a constraint optimization problem, outputting the desired ground reaction force or joint torque. It has been successfully applied to robot platforms such as Mini-Cheetah and Anymal, supporting high-performance movements such as dynamic jumps and somersaults. However, MPC is extremely sensitive to model accuracy; system parameter errors or external disturbances can significantly affect its control accuracy and stability.

[0005] To enhance the robustness of MPC, existing research has attempted to combine it with strategies such as robust control and adaptive control. For example, robust MPC enhances control stability within known disturbance boundaries, but often sacrifices system performance due to conservative design; adaptive MPC can improve system adaptability through online parameter estimation, but it is insufficient for adapting to drastic disturbance changes. Disturbance observers (DOBs), as a typical disturbance compensation mechanism, have also been introduced into MPC to improve system disturbance immunity, but their estimation accuracy usually depends on the assumption of bounded disturbance derivatives, making it difficult to cope with rapidly changing or high-order nonlinear disturbances.

[0006] Therefore, this invention proposes a model predictive control method for quadruped robots that integrates an extended state observer.

[0007] In this invention, the Extended State Observer (ESO) is a model-independent observer capable of real-time estimation of system state and total disturbance, offering advantages such as simple structure, high estimation accuracy, and low computational cost. Integrating the ESO into the MPC control framework promises to achieve accurate compensation for unknown disturbances without increasing sensor load, thereby improving the motion robustness and control performance of quadruped robots in uncertain dynamic environments. Therefore, the ESO-based model predictive control method provides a new and effective approach to address the bottlenecks in disturbance estimation and compensation in existing MPC control. Summary of the Invention

[0008] The purpose of this invention is to provide a model predictive control method for quadruped robots that integrates an extended state observer. This method addresses the problems of decreased motion control accuracy and insufficient disturbance rejection capability in quadruped robots operating in complex dynamic environments, particularly the performance degradation of existing Model Predictive Control (MPC) methods when faced with external unknown disturbances and model uncertainties. Existing methods based on Disturbance Observer (DOB) often rely on the assumption of bounded disturbance derivatives, making it difficult to accurately estimate higher-order non-stationary disturbances, resulting in incomplete compensation and insufficient stability.

[0009] The specific technical solution adopted by this invention is as follows:

[0010] Based on the characteristic that Extended State Observer (ESO) can estimate the total disturbance without requiring a known disturbance model, this invention proposes a model predictive control method for a quadruped robot under uncertain dynamic environments based on ESO. This method employs a dual ESO structure to estimate unknown disturbances in the translational and rotational directions separately, and feeds these estimates forward into a discrete MPC framework to improve the system's disturbance rejection capability and trajectory tracking accuracy. The method includes the following steps:

[0011] The method establishes a single rigid body dynamic model of the quadruped robot with the center of mass as the reference, and models the unknown perturbation force and perturbation torque on the body as a unified system perturbation term, forming a state space model with perturbation, which provides a basis for the subsequent design of observers and controllers.

[0012] The extended state observer of the method is designed with multi-level position ESO and attitude ESO respectively;

[0013] The position ESO is used to estimate the external disturbance forces on the robot in the x, y, and z directions;

[0014] The attitude ESO is used to estimate the disturbance torques about the roll axis, pitch axis, and yaw axis.

[0015] In the observer design, an appropriate bandwidth parameter is selected to ensure that the perturbation estimation error is bounded, and the stability and convergence are analyzed using the linear system method;

[0016] The proposed method constructs the MPC state space by introducing the disturbance term estimated by ESO as a compensation quantity into the discrete state space model, thereby enhancing the robustness of the controller without increasing the burden on additional sensors.

[0017] The MPC optimization problem of the method is specifically defined as a cost function with the goal of minimizing the expected trajectory tracking error. Combining the system's operational safety and the physical constraints of the actuator, the following optimization constraints are constructed: the ground reaction force satisfies the friction cone constraint; the contact force of the swing leg is zero; the foot force and joint torque are limited by the upper and lower limits of the system; and the state variables must satisfy the system's physical boundary conditions.

[0018] The MPC rolling optimization solution of the method specifically adopts a rolling time-domain control strategy, which solves the finite time-domain optimal control problem with disturbance compensation in real time in each control cycle to obtain the desired foot force or joint torque, and uses it as the controller output to act on the quadruped robot.

[0019] The simulation verification of the method specifically involves conducting a trajectory tracking experiment on a Unitree Go1 quadruped robot using the Gazebo simulation platform on the Ubuntu 20.04 system. This verifies that when carrying unmodeled disturbance loads, the method of the present invention significantly improves performance indicators such as body height error, desired speed error, and attitude fluctuation compared to the traditional DOB-MPC controller, demonstrating stronger anti-disturbance capability and control stability.

[0020] The technical effects achieved by this invention are as follows:

[0021] This invention constructs a dual ESO structure to achieve real-time estimation of unknown disturbance forces and moments, and then incorporates this estimation into the MPC controller to achieve high-precision trajectory tracking and enhanced system robustness. It effectively overcomes the problems of poor estimation accuracy and strong model dependence of traditional disturbance observation methods, and possesses advantages such as simple structure, good real-time performance, and strong disturbance resistance.

[0022] This invention can be widely applied to high-precision motion control tasks of various quadruped robot systems in uncertain environments such as complex terrain, load changes, and unknown disturbances. It is suitable for application scenarios such as search and rescue, field inspection, and intelligent transportation, and has good engineering practicality and promotion value. Attached Figure Description

[0023] Figure 1 This is a model predictive control framework diagram based on an extended state observer, according to the present invention.

[0024] Figure 2 This is a parameter diagram of the Go1 of this invention carrying a 6kg payload in a simulation environment;

[0025] Figure 3 This is a diagram showing the rotation parameters of the Go1 of this invention carrying a 6kg effective load in a simulation environment. Detailed Implementation

[0026] To make the objectives and advantages of this invention clearer, the invention will be specifically described below with reference to embodiments. It should be understood that the following text is merely used to describe one or more specific embodiments of the invention and does not strictly limit the scope of protection specifically claimed by the invention.

[0027] like Figure 1 As shown, a predictive control method for a quadruped robot model that integrates an extended state observer is characterized by the following steps:

[0028] Step 1: Construct a dynamic model of the quadruped robot, establish the kinematic equations of a single rigid body with the center of mass as the reference point, introduce system perturbation terms, and form a state-space model with perturbation.

[0029] The dynamic model of a quadruped robot essentially describes the dynamic coupling relationship between the system's internal forces and changes in its motion state. To achieve the robot's expected motion trajectory, it is necessary to calculate the dynamic balance between the driving joint torques and the foot contact forces. This invention establishes a quantitative mapping between the foot forces and the body's motion parameters by constructing mathematical-physical equations, providing core algorithmic support for subsequent control algorithms.

[0030] In this invention, the robot kinematics modeling method in step 1 specifically includes:

[0031] To describe the overall dynamic behavior of a quadruped robot under disturbance, a single rigid body kinematics modeling approach with the center of mass as the reference point is adopted. This model ignores the influence of the flexibility of the leg joints and mainly characterizes the six-degree-of-freedom rigid body dynamics of the robot body. It has the advantages of simple modeling and easy controller design. Its representation is as follows:

[0032]

[0033] in, For the quality of the robot; Let be the rotational inertia of the robot's center of mass; It is the gravity vector; The base angular velocity of the robot; The location of the center of mass; i = 1, 2, 3, 4 represents the position of the foot; For the contact force of each leg;

[0034] Considering the impact of unknown disturbances on the robot's motion, a single rigid body dynamics model incorporating the disturbance term is established as follows:

[0035]

[0036] in, This indicates the uncertainty of the robot in the translational direction. This represents the uncertainty in the robot's rotational direction; in this case, if ω is ignored... b ×I G ω b Due to the influence of the above formula, it can be expanded into the following form.

[0037]

[0038] It is worth noting that this approximation method is widely used in the motion control of quadruped robots, and the approximation is considered reasonable, especially when the angular velocity of the object is small, ω b ×I G ω b Negligible;

[0039] To complete the observer design, the following assumptions are proposed:

[0040] Assumption 1.F b d(t), d(t), and their time derivatives have upper bounds.

[0041] That is, ||F b (t)‖2≤γ1,‖d(t)‖2≤γ2, and Where γ1, γ2, δ1 and δ2 are random positive constants.

[0042] Step 2: Design the dual extended state observer (ESO), including:

[0043] Position ESO is used to estimate unknown disturbance forces in the robot's translational direction in real time;

[0044] Attitude ESO is used to estimate disturbance moments about the roll, pitch, and yaw axes in real time.

[0045] Based on the ESO knowledge described in step S1, the design steps for the extended state observer of the quadruped robot can be divided into two steps: extended state observer design based on disturbance force and extended state observer design based on disturbance torque. The specific design process is as follows:

[0046] The ESO nonlinear control method is used to estimate the unknown disturbance force F acting on the robot. b And the uncertain torque d. After estimating the two disturbance terms, they are incorporated into the MPC framework to ensure the friction constraint of the ground reaction force. Finally, the desired foot force and the correction force generated by the swing leg dynamics are optimized and converted into the joint output torque τ. d .

[0047] Define an input Output The system, in the form of u (n) =Δ+x, where, This is an unknown term in the system. The output u can be measured by a sensor. Let u1 = u, u n =u (n) Define u n+1 To extend the state variable, and u n+1 =Δ. The original controlled object is now described as

[0048] The purpose of ESO is to estimate Δ using a measurable output u. Let Let u1, ..., u n+1 Given the estimated value, the ESO design is as follows:

[0049]

[0050] Where, a1=w0α1, α i = (n+1)! / [i! (n+1-i)!], i = 1, 2, ..., n+1. Let ω0 be the observer bandwidth, and ω0 > 0. Let... (i = 1, ..., n+1) represents the estimation error. Combining this with formula (1.33), the error dynamics expression is:

[0051]

[0052] Stability analysis shows that if Bounded, that is γ o If r is a random constant, then there exists a constant r. i >0 and a finite time T1>0, such that For all t≥T1≥0 and ω0>0.

[0053] Set constant r i as follows:

[0054]

[0055] Where k is a positive integer, h i Since Δ is a constant, it can be seen from formula (1.35) that the estimation error bound of ESO is determined by the initial estimation error and the observer bandwidth ω0; since the unknown term Δ cannot be measured, the initial estimation error and γ are constants. o It cannot be used to reduce r i At the same time, the gain ω0 and It is inversely proportional; therefore, when the value of ω0 in ESO is large, a smaller r can be obtained. i value;

[0056] Based on the above ESO knowledge, the design steps of the extended state observer for a quadruped robot can be divided into two steps: extended state observer design based on disturbance force and extended state observer design based on disturbance torque. The specific design process is shown below:

[0057] Step 201: The design of the extended state observer based on disturbance force in step 2 specifically includes:

[0058] According to formula (1.2):

[0059]

[0060] in, To control input items; This refers to the positional disturbance force acting on the robot's center of mass.

[0061] Design an n-order position ESO to estimate the unknown term. make Let p, v, and f represent respectively. Δ,1 ,...,f Δ,n The estimate; then the nth-order ESO design is:

[0062]

[0063] in, They represent The i-th element; x v,i For x v The i-th element; For ESO adjustable parameters; according to ESO design and formula (1.3), the estimated error is bounded, i.e. From formula (1.3), we can see that r p,1 ,r p,2 ,r p,3 With ω p,1 ,ω p,2 ,ω p,3 It is inversely proportional to the k-th power;

[0064] At the same time, use a large gain vector ω p To obtain a smaller estimation error bound, we can use the position of ESO; at this point, let r p Indicates r p,1 ,r p,2 ,r p,3 Three times the maximum value, i.e., r p =3max{r p,1 ,r p,2 ,r p,3 If}, then there exists a limit.

[0065] The closed-loop dynamics of the position loop are divided into two sub-loops: position and velocity. This dual-sub-loop structure is matched to the cascaded structure of a quadruped robot. Let the desired position and position error of the base be represented respectively. Then the expected dynamics of the two sub-loops are as follows:

[0066]

[0067] in, It is a diagonal matrix; the position error is defined as s. v =v c -v r v c v is the actual speed. r The desired velocity reference trajectory; to achieve the desired position sub-loop, v r Designed as follows:

[0068]

[0069] v r and The relationship is shown in formula (1.24); at the same time, the velocity error s is obtained. v The relationship with position error is as follows:

[0070]

[0071] As shown in formula (1.8), the desired positional sub-loop can be obtained through r. v →0 is used to achieve this; in order to illustrate s v and To understand the relationship between position dynamics and the stability of position dynamics, we introduce the following technical lemma 1, assumption 2, and lemma 2:

[0072] Lemma 1: Let r1 denote a constant, if ||s v If ||≤r1, t→∞, then t→∞, where, express traces;

[0073] Assumption 2: Assumption f Δ (n) If it is bounded, then we can obtain Bounded, that is

[0074] Lemma 2: According to Assumption 2, if the position control law is as shown in Formula (1.9) and the position ESO is as shown in Formula (1.5), then the position tracking error is... Bounded, that is t→∞, where r p This is the upper bound of the position ESO estimation error;

[0075]

[0076] Step 202: The design of the disturbance-based extended state observer in step 2 specifically includes:

[0077] According to formulas (1.2) and (1.3), the dynamic model can be rewritten as:

[0078]

[0079] in, For control input; For disturbance torque;

[0080] Design an n-order attitude ESO to estimate unknown disturbance terms. make They represent ω respectively b ,τ Δ,1 ,...,τ Δ,n The estimation of the nth-order attitude ESO is then performed as follows:

[0081]

[0082] Where, x ω,i It is x ω The i-th element, For the attitude ESO adjustable bandwidth vector, They are respectively and The i-th element; as mentioned above, the estimation error is bounded, i.e. And because r ω,1 ,r ω,2 ,r ω,3 With ω ω,1 ,ω ω,2 ,ω ω,3 The value is inversely proportional to the k-th power of the gain vector; therefore, the position of the ESO with a larger gain vector obtains a smaller estimation error bound; let r ω Indicates r ω,1 ,r ω,2 ,r ω,3 Three times the maximum value, i.e., r ω =3max{r ω,1 ,r ω,2 ,r ω,3 If}, then there exists a limit. Let ω d The desired angular velocity of the quadruped robot is represented by the following formula:

[0083]

[0084] Define the error rotation matrix as Define error quaternions in, and

[0085]

[0086] set up The derivative of the error rotation matrix can be obtained using the following formula:

[0087]

[0088] in,[·] × This represents the cross product operation;

[0089] To achieve motion control of the quadruped robot, the closed-loop dynamics of the attitude loop are divided into two sub-loops: attitude and angular velocity. The design dynamics of the two sub-loops are given below:

[0090]

[0091] in, and Let k be a diagonal matrix. β >0 represents a constant gain; furthermore, s ω =ω-ω r ω r Provide a reference trajectory for the required angular velocity;

[0092] To achieve the desired attitude sub-loop, ω r Designed as follows:

[0093]

[0094] roll out

[0095]

[0096] The desired attitude sub-loop is then s. ω →0;

[0097] To achieve the desired angular velocity sub-loop, the quadruped robot base attitude controller is designed as follows:

[0098]

[0099] in,

[0100]

[0101] Substituting the control law (1.19) into formula (1.10) yields:

[0102]

[0103] according to Bounded; under control law (1.19), the desired angular velocity subloop is realized in a bounded manner, then the control torque vector T B for:

[0104]

[0105] It is worth noting here that, in order to illustrate the stability of the attitude loop, we introduce the following assumption 3, Lemma 3;

[0106] Assumption 3: Assumption If it is bounded, then it can be obtained. Bounded, that is

[0107] Lemma 3: According to Assumption 3, if the attitude controller is designed as Equation (1.21) and the attitude ESO is Equation (1.11), then the attitude tracking error is bounded, i.e. t→∞, where r q This is the upper bound of the attitude tracking error.

[0108] Step 3: The disturbance estimate output by ESO is used as a feedforward compensation term and embedded into the constructed discrete-time state-space model to form a model predictive control (MPC) system with disturbance compensation.

[0109] This step takes into account the design of the single rigid body dynamics model and the extended state observer mentioned above. To facilitate controller design, this invention further constructs a state-space representation of the quadruped robot system for subsequent model prediction controller optimization modeling.

[0110] The construction of the model predictive control system with disturbance compensation in step 3 specifically includes:

[0111] According to formulas (1.4) and (1.16), the continuous-time state-space model of the quadruped robot system is expressed as:

[0112]

[0113] Its discrete-time state-space model can be represented as:

[0114]

[0115] The variables in formula (1.23) are defined as follows:

[0116] State variables for:

[0117]

[0118] in, The location of the robot's center of mass; Attitude angles; such as Euler angles; The linear velocity of the center of mass; The angular velocity of the machine body; The disturbance force and disturbance moment are estimated from the position ESO and attitude ESO.

[0119] Input control The expression for the expected ground reaction force of a quadruped robot is:

[0120]

[0121] in, Let it be expressed as the expected ground reaction force of the nth leg;

[0122] Gravity term vector The expression for gravitational acceleration is:

[0123]

[0124] ESO estimator vector The ESO estimate containing the interference term and its derivative, used for feedforward compensation, are expressed as follows:

[0125]

[0126] Transition matrix The fundamental dynamic updates of the system, including position and velocity integrals and attitude updates, are expressed as follows:

[0127]

[0128] Input matrix The expression is:

[0129]

[0130] Where m is the robot mass, I G For the body's inertia matrix, [p i -p c ] × Δt, i = 1, 2, 3, 4 is the position vector of the foot relative to the center of mass;

[0131] Disturbance compensation matrix The expression for incorporating the ESO-estimated disturbance term into the discrete model to achieve feedforward disturbance compensation is as follows:

[0132]

[0133] The above describes the detailed implementation method of the discrete state-space model construction in this invention. By modeling the system state, control input, and disturbance compensation terms in a unified matrix form, a rigorous and programmable numerical foundation is provided for the subsequent solution of the MPC optimizer. This form features clear structure, explicit physical meaning of parameters, and strong scalability, making it suitable for unified implementation in simulation and actual controller deployment.

[0134] After constructing a discrete state-space model including disturbance compensation terms, this invention formulates a finite-time linear model predictive control (MPC) problem based on this model. The prediction range length is n, that is, the optimizer predicts the system behavior for the next n steps in each control cycle and solves the optimal control input sequence online. This linear MPC problem can be expressed as follows:

[0135]

[0136] in, This represents the system state at time i. This represents the desired state of the system at any given time. F i The ground reaction force R is calculated at time step i. i and S i It is a diagonal positive semi-definite matrix. The components of the contact force vector for each foot are represented by F. ix F iy and F izFurthermore, μ represents the coefficient of friction between the contact point and the ground, K i It is the inequality constraint matrix, D i It is the force constraint matrix of the swing leg.

[0137] The linear MPC problem constructed in this invention combines an ESO-based real-time perturbation estimation mechanism, effectively overcoming the problems of traditional MPC's strong dependence on model accuracy and insufficient robustness to perturbations. This optimized model has the following advantages: accurate state prediction, adaptability to complex trajectory changes, perturbation feedforward compensation to improve disturbance rejection capability, support for multiple constraints to adapt to actual robot physical limitations, and a clear algorithm structure that facilitates engineering implementation and real-time deployment. Furthermore, this controller structure has been verified on simulation platforms and in actual quadruped robot systems, demonstrating excellent trajectory tracking performance and robustness. The specific verification methods are described in step S5.

[0138] Step S4: Implementation of Experimental Verification and Analysis

[0139] To verify the effectiveness and robustness of the quadruped robot model predictive control method (ESO-MPC) with integrated extended state observer proposed in this invention, this step designed a simulation experiment to test it on a virtual loop, evaluate its motion control performance under complex disturbance conditions, and compare it with the traditional disturbance observer combined with MPC method (DOB-MPC).

[0140] This invention establishes an experimental environment on the Gazebo simulation platform, using Ubuntu 20.04 as the operating system. The URDF model description file of the Unitree Go1 quadruped robot is used in the experiment to ensure consistency between the simulation environment and the real platform in terms of structure and parameters. The controller is implemented in C++ and uses a control interface under the ROS framework to achieve real-time driving and interaction with the simulated robot.

[0141] This experiment aims to evaluate the Go1 robot's ability to maintain desired center-of-gravity height, linear velocity, and angular velocity under perturbation conditions when carrying a payload of unknown mass. The proposed ESO-MPC control method and the traditional DOB-MPC method were used to perform linear velocity tracking on the robot. The actual motion trajectory of the robot under different control strategies was compared with the set reference trajectory to verify the effectiveness and robustness of the proposed control framework.

[0142] In the simulation experiments of this invention, the following two typical simulation experiment scenarios were designed:

[0143] Scenario 1: Straight-line walking experiment carrying a 6kg payload

[0144] In this experiment, the robot's center of mass (CoM) height was set to 0.32m, and the desired forward speed was set to 1.5m / s. The high-speed linear motion of the robot under constant load disturbance was simulated in a simulation platform. The proposed ESO-MPC control method and the traditional DOB-MPC control method were used for control, and the actual tracking curves of CoM height and CoM speed were recorded. The simulation comparison results are as follows: Figure 2 As shown.

[0145] Experimental data show that both control methods can achieve relatively accurate centroid velocity tracking, exhibiting good disturbance rejection and stability. Among them, the ESO-MPC method is slightly superior to the DOB-MPC method in tracking error control, demonstrating higher velocity tracking accuracy and control smoothness, thus verifying the control advantages of the method proposed in this invention when dealing with payload disturbances.

[0146] Scenario 2: In-situ rotation experiment carrying a 6kg payload

[0147] In this experiment, the robot's center of mass height was set to 0.32m, and the desired rotational angular velocity was set to 2rad / s. The robot performed a stationary rotation task without moving its position, and the CoM height and base angular velocity were recorded to test its attitude tracking capability and disturbance torque compensation capability. The simulation comparison results are as follows: Figure 3 As shown.

[0148] Experimental results show that both methods can achieve relatively stable angular velocity output in stationary rotation control tasks. Among them, the ESO-MPC control method has a faster response and smaller oscillation in disturbance compensation, and exhibits higher attitude control stability and robustness.

[0149] Comparison of Error Analysis and Quantitative Results

[0150] Based on the fuselage height tracking errors e1 and e2 recorded during the simulation, their L2 norms were calculated, and the results are shown in Table 1. Experimental data show that, under the same interference conditions and desired input, the ESO-MPC method outperforms the DOB-MPC method in terms of fuselage height tracking error.

[0151]

[0152] Table 1 shows the L2 norms of the tracking errors e1 and e2 of the fuselage height during translation and rotation in the simulation experiment.

[0153] The above description is merely a preferred embodiment of the present invention. It should be noted that those skilled in the art can make various improvements and modifications without departing from the principles of the present invention, and these improvements and modifications should also be considered within the scope of protection of the present invention. Structures, devices, and operating methods not specifically described or explained in this invention are implemented according to conventional methods in the art unless otherwise specified or limited.

Claims

1. A predictive control method for a quadruped robot model incorporating an extended state observer, characterized in that: It includes the following steps: Step 1: Construct a dynamic model of the quadruped robot, establish the kinematic equations of a single rigid body with the center of mass as the reference point, introduce system perturbation terms, and form a state-space model with perturbation. Step 2: Design the dual extended state observer (ESO), including: Position ESO is used to estimate unknown disturbance forces in the robot's translational direction in real time; Attitude ESO is used to estimate disturbance moments about the roll, pitch, and yaw axes in real time. Step 3: The disturbance estimate output by ESO is used as a feedforward compensation term and embedded into the constructed discrete-time state-space model to form a model predictive control (MPC) system with disturbance compensation. ESO estimator vector The ESO estimate containing the interference term and its derivative, used for feedforward compensation, are expressed as follows: （1.27） in, The disturbance force and disturbance moment are estimated from the position ESO and attitude ESO. Step 4: Construct a finite-time linear MPC optimization problem, using the minimum trajectory tracking error as the cost function, while considering the following constraints: the ground reaction force satisfies the friction cone constraint; the contact force of the swing leg is zero; the foot force and joint torque are within the upper and lower limits; and the state variables satisfy the physical boundary conditions. Step 5: Solve the MPC problem online using the rolling optimization method to obtain the optimal foot reaction force and joint torque, and apply them to the quadruped robot system to complete robust trajectory tracking control under disturbance environment; Specifically, the construction and solution of the MPC optimization problem include: After constructing a discrete state-space model including disturbance compensation terms, a finite-time linear model predictive control (MPC) problem is formulated based on this model, with a prediction range length of... That is, the optimizer predicts the future within each control cycle. The system behavior is analyzed step by step, and the optimal control input sequence is solved online. This linear MPC problem can be formulated as follows: （1.31） （1.32） in, Indicates time The system state at any given moment. This represents the desired state of the system at any given time. In time steps Calculated ground reaction force, and It is a diagonal positive semi-definite matrix; the components of the contact force vector for each foot are expressed as follows: and ;also, This represents the coefficient of friction between the contact surface and the ground. It is an inequality constraint matrix. It is the force constraint matrix of the swing leg.

2. The quadruped robot model predictive control method according to claim 1, characterized in that, The robot kinematics modeling method in step 1 specifically includes: The kinematic modeling method using a single rigid body with the center of mass as the reference point is adopted; its representation is as follows: （1.1） in, For the quality of the robot; Let be the rotational inertia of the robot's center of mass; It is the gravity vector; The base angular velocity of the robot; The location of the center of mass; The foot position; , For the contact force of each leg; The single rigid body dynamic model incorporating disturbance terms is established as follows: （1.2） in, This indicates the uncertainty of the robot in the translational direction. This represents the uncertainty in the robot's rotation direction; in this case, if we ignore... Due to the influence of the above formula, it can be expanded into the following form. （1.3） When the angular velocity of an object is small, Negligible; To complete the observer design, the following assumptions are proposed: Assumption 1. Its time derivative has an upper bound. Right now and ,in, and It is a random positive number.

3. The predictive control method for a quadruped robot model with an integrated extended state observer according to claim 2, characterized in that, The design of the extended state observer based on disturbance force in step 2 specifically includes: According to formula (1.2): （1.4） in, To control input items; This refers to the positional disturbance force acting on the robot's center of mass. design ESO estimation of unknown terms at order position ,make They represent The estimate; then The ESO design is as follows: （1.5） in, They represent The One element; for The One element; These are adjustable parameters for the ESO; according to the ESO design and formula (1.3), the estimated error is bounded, i.e. Furthermore, as can be seen from formula (1.3), and of The power is inversely proportional to the power; At the same time, use a vector with a large gain. To obtain a smaller estimation error bound, we use the position of ESO; at this time, let express Three times the maximum value, that is Then there exists a limit. ; The closed-loop dynamics of the position loop are divided into two sub-loops: position and velocity. This dual-sub-loop structure is matched to the cascaded structure of a quadruped robot. Let the desired position and position error of the base be represented respectively. Then the expected dynamics of the two sub-loops are as follows: （1.6） in, It is a diagonal matrix; the position error is defined as... For actual speed, The desired velocity reference trajectory; in order to achieve the desired position sub-loop, Designed as follows: （1.7） and The relationship is shown in formula (1.24); at the same time, the velocity error is obtained. The relationship with position error is as follows: （1.8） As can be seen from formula (1.8), the desired position sub-loop can be obtained through... To achieve; in order to explain and To understand the relationship between positions and the stability of position dynamics, we introduce the following technical lemma 1, assumption 2, and lemma 2: Lemma 1: Let Denote a constant, if ,but ,in, express traces; Assumption 2: Assumption If it is bounded, then we can obtain Bounded, that is ; Lemma 2: According to Assumption 2, if the position control law is as shown in Formula (1.9) and the position ESO is as shown in Formula (1.5), then the position tracking error is... Bounded, that is ,in, This is the upper bound of the position ESO estimation error; （1.9）。 4. The predictive control method for a quadruped robot model with an integrated extended state observer according to claim 3, characterized in that, The design of the disturbance-based extended state observer in step 2 specifically includes: According to formulas (1.2) and (1.3), the dynamic model can be rewritten as: （1.10） in, For control input; For disturbance torque; design ESO estimation of unknown disturbance terms ,make They represent The estimate; then The step attitude ESO design is as follows: （1.11） in, yes The One element, For the attitude ESO adjustable bandwidth vector, They are respectively and The There are 10 elements; as mentioned above, the estimation error is bounded, i.e. Furthermore, due to and of The power is inversely proportional; therefore, the position of ESO with a larger gain vector obtains a smaller estimation error bound; let express Three times the maximum value, that is Then there exists a limit. ;make The desired angular velocity of the quadruped robot is represented by the following formula: （1.12） Define the error rotation matrix as Define error quaternions ,in, ,and （1.13） set up The derivative of the error rotation matrix can be obtained using the following formula: （1.14） in, This represents the cross product operation; To achieve motion control of the quadruped robot, the closed-loop dynamics of the attitude loop are divided into two sub-loops: attitude and angular velocity. The design dynamics of the two sub-loops are given below: （1.15） in, and It is a diagonal matrix. It is a constant gain; in addition, Provide a reference trajectory for the required angular velocity; To achieve the desired attitude sub-loop, Designed as follows: （1.16） roll out （1.17） The desired attitude sub-loop is then achieved as follows: ; To achieve the desired angular velocity sub-loop, the quadruped robot base attitude controller is designed as follows: （1.18） in, （1.19） Substituting the control law (1.19) into formula (1.10) yields: （1.20） according to Bounded; under the control law (1.19), the desired angular velocity subloop is realized in a bounded manner, then the control torque vector for: （1.21） To illustrate the stability of the attitude loop, we introduce the following assumption 3, Lemma 3; Assumption 3: Assumption If it is bounded, then it can be obtained. Bounded, that is ; Lemma 3: According to Assumption 3, if the attitude controller is designed as Equation (1.21) and the attitude ESO is Equation (1.11), then the attitude tracking error is bounded, i.e. , where is the upper bound of the attitude tracking error.

5. The quadruped robot model predictive control method according to claim 4, characterized in that, The construction of the model predictive control system with disturbance compensation in step 3 specifically includes: According to formulas (1.4) and (1.16), the continuous-time state-space model of the quadruped robot system is expressed as: （1.22） Its discrete-time state-space model can be represented as: （1.23） The variables in formula (1.23) are defined as follows: State variables for: （1.24） in, The location of the robot's center of mass; For attitude angle; The linear velocity of the center of mass; The angular velocity of the machine body; The disturbance force and disturbance moment are estimated from the position ESO and attitude ESO. Input control The expression for the expected ground reaction force of a quadruped robot is: （1.25） in, Represented as the first The expected ground reaction force of a single leg; Gravity term vector The expression for gravitational acceleration is: （1.26） ESO estimator vector The ESO estimate containing the interference term and its derivative, used for feedforward compensation, are expressed as follows: （1.27） Transition matrix The fundamental dynamics update of the system, including position-velocity integrals and attitude updates, is expressed as: （1.28） Input matrix The expression is: （1.29） in, For robot quality, For the body's inertial matrix, The position vector of the foot relative to the center of mass; Disturbance compensation matrix This is used to introduce the disturbance term estimated by ESO into the discrete model to achieve feedforward disturbance compensation. The expression is: （1.30）。

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