Gait control method for 12-degree-of-freedom biped humanoid robot

Abstract

The invention discloses a gait control method for a 12-degree-of-freedom biped humanoid robot. The gait control method comprises the following steps that S1, structural design of a self-research robot is conducted; s2, modeling the self-developed biped humanoid robot; s3, designing a self-developed biped humanoid robot controller; s4, performing evaluation and experimental analysis; according to the robot, six joint drivers are arranged for each leg at three main parts of a hip joint, a knee joint and an ankle joint, sufficient hardware conditions are provided for flexible movement of a self-developed robot, a humanoid robot model is simplified by adopting a single rigid body model, and a robot movement control method and constraint conditions thereof are designed for 12 degrees of freedom, so that the robot movement control method and constraint conditions are optimized. Although the simplified model ignores leg dynamics, robust stability of a 12-degree-of-freedom system can still be achieved under the multi-body effect, and adaptability and real-time performance of the system are improved.

Description

Technical Field

[0001] The present invention belongs to the field of robot control, and in particular relates to a gait control method for a bipedal humanoid robot with 12 degrees of freedom. Background Art

[0002] Bipedal humanoid robots have attracted widespread attention due to their superior mobility over traditional robots on complex terrain, potentially replacing humans in hazardous work environments. A moving bipedal humanoid robot is only affected by gravity and ground reaction forces. However, these forces cannot be directly controlled; the robot's motion can only be adjusted by controlling the driving forces of its joints, which complicates its control. Furthermore, when walking or running on complex terrain, bipedal humanoid robots must maintain balance and move safely even in the presence of external disturbances. For medium- to large-sized humanoid robots with 12 degrees of freedom (DOF), motion control presents challenges not only in terms of stability but also in terms of real-time performance. Therefore, designing an effective control method to achieve high dynamic stability in complex terrain is crucial for bipedal humanoid robots with 12 degrees of freedom (DOF).

[0003] Numerous approaches have been developed to control bipedal humanoid robots. The zero moment point (ZMP) approach is a representative approach for bipedal humanoid robots. It simplifies the robot's dynamics into an inverted pendulum model and achieves stability during motion by minimizing the error between the actual ZMP position and a reference ZMP position. Typically, the ZMP requires that the center of mass maintain its height during motion, so the ZMP approach requires the robot to walk with bent knees. The spring-loaded inverted pendulum model simplifies the robot's dynamics into a spring-loaded inverted pendulum, decoupling the complex control into height control, forward velocity control, and body posture control. Hybrid zero dynamics is another control framework based on whole-body dynamics. It uses input-output linearization to design a nonlinear feedback controller with virtual constraints to generate a stable walking gait. This approach is suitable for underactuated bipedal humanoid robots, but the gait can be unstable due to large disturbances. Furthermore, reinforcement learning has been applied to the dynamic motion control of quadrupedal robots. However, these approaches are unable to handle state constraints due to actuator limitations and friction constraints.

[0004] Model predictive control (MPC) is capable of handling systems with constraints, multiple variables, and nonlinearities, while flexibly adapting to different control objectives and providing more precise and stable control by predicting changes in system behavior over a period of time. In recent years, MPC has been widely used in the control of bipedal humanoid robots. Therefore, the present invention proposes a control framework based on MPC to achieve force and torque control of a 12-DOF bipedal humanoid robot. The constraints based on MPC are designed mainly for the control of a 12-DOF bipedal robot. Due to the design of force and torque-based control and the consideration of stability constraints, the proposed MPC controller is robust to complex terrain during dynamic motion. Summary of the Invention

[0005] The purpose of the present invention is to provide a gait control method for a bipedal humanoid robot with 12 degrees of freedom. The extended Kalman filter technology is innovatively used to give the best estimate in the maximum likelihood sense for the data measured by the infrared sensor, thereby reducing the impact of noise on the system.

[0006] To solve the above technical problems, the technical solution adopted by the present invention is: a gait control method for a 12-degree-of-freedom bipedal humanoid robot, comprising the following steps:

[0007] Step S1: Structural design of the self-developed robot; the self-developed humanoid robot's legs are designed with 12 degrees of freedom to achieve flexible motion control. Each leg of the robot is equipped with six joint drive mechanisms, and the joints are distributed in three main parts: the hip, knee, and ankle, forming a complex motion system. The hip joint consists of three joints: hip roll, yaw, and pitch, which are responsible for the robot's rolling, yaw, and tilting movements in space respectively; the knee joint is equipped with a pitch joint, which is mainly responsible for leg flexion and extension; the ankle joint contains two joints: roll and pitch, which are responsible for the flexible adjustment of the robot's foot to adapt to different terrains and postures. The overall structure of the robot consists of a body link, a hip roll link, a hip yaw link, a knee pitch link, an ankle roll link, and an ankle pitch link. These links support the robot's complex rotation and movement in three-dimensional space and can maintain balance and stability while supporting the ground with both legs.

[0008] Step S2: Model the self-developed bipedal humanoid robot. Because the full-body dynamics of the bipedal humanoid robot are highly nonlinear and complex, a single rigid-body model is used for simplification to ensure the real-time performance of the model predictive control controller. Although this simplified model ignores the leg dynamics, the model predictive control-based controller still achieves robust stability for the high-degree-of-freedom system under multibody effects. The single rigid-body model is represented in the world coordinate system as follows:

[0009]

[0010] Represents the position coordinates of the center of mass, p x is the position of the center of mass along the x-axis, p y is the position of the center of mass along the y-axis, p z is the position of the center of mass along the z axis, m is the mass of the robot, and They represent the ground contact force and torque of leg i (i=1,2) in the world coordinate system, i=1 corresponds to the right leg of the robot, i=2 corresponds to the left leg, and are the longitudinal force, lateral force and vertical force respectively. xi , F yi , and F zi are longitudinal force, lateral force and vertical force respectively, vector g=[0 0 g] T , where g is the acceleration due to gravity, and are the robot angular velocity and moment of inertia in the world coordinate system, is the distance vector from the robot's center of mass to the foot contact point.

[0011] Step S3: Design a self-developed bipedal humanoid robot controller. To implement an MPC controller based on a linearized state-space dynamics model, the forward Euler method is used to discretize the robot's dynamics state-space equation at the kth time step, yielding:

[0012] x(k+1)=A k x(k)+B k ku(k).

[0013] in

[0014] A k =I 13 +AT s , B k =BT s ;

[0015] Based on the discrete-time model, a model predictive control problem with a finite time domain h is constructed, and its quadratic cost function is expressed as follows:

[0016]

[0017] The constraints included are:

[0018] x(k+1)=A k x(k)+B k u(k)

[0019] -μF zi ≤F xi ≤μFzi

[0020] -μF zi ≤F yi ≤μF zi

[0021] bF z，min ≤F zi ≤μF z，max

[0022] M x =0

[0023] -F z ·l t ≤M y ≤F z ·l h

[0024] F y ·l t -μ·M y -μ·l t ·F z ≤M z ≤μ·M y +μ·l t ·F z -F y ·l t

[0025] μ·l h ·F z -μ·M y -F y ·l h ≤M z ≤μ·M y +F y ·l h -μ·l h ·F z

[0026] The first term is used to track the desired state x r , the second one aims to minimize the control input to reduce the energy consumption of movement; Q k and R k is a diagonal weight matrix, which is used to adjust the dynamic response characteristics of the state variables and control inputs respectively. The control problem needs to meet the following constraints: (1) discrete dynamics constraint; (2) contact point friction constraint; (3) line foot dynamics constraint to ensure that the robot meets the contact torque stability criterion. Among the specific parameters, μ represents the ground friction coefficient, F z，min and F z，max Represent the minimum and maximum thresholds of vertical force, l h and l tThey represent the distance from the foot contact point to the heel and toes respectively.

[0027] To ensure the real-time performance of model predictive control, the control problem represented by the quadratic cost function can be transformed into a quadratic programming problem. Specifically, the discrete-time dynamic model with a finite time domain h can be expressed as:

[0028] X=A q x(0)+B q U;

[0029] Where the state vector X=[x(1) T x(2) T … x(h) T ] T ;

[0030] Input vector U=[u(0) T u(1) τ … u(h-1) T )] T , x(0) represents the initial state of the system, matrix A q and B q A containing h time domains in the future k and B k Matrix. From this, we can get the following quadratic programming problem:

[0031]

[0032] The constraints are:

[0033] A i U≤B i , A e U=B e ;

[0034] Among them, A i and B i is the inequality constraint matrix, A e and B e is the equality constraint matrix, and the matrices H and f can be expressed as:

[0035]

[0036] Among them, X r is a column vector containing the expected system state in the future h time domains, Q and R are the weight matrices Q containing the future h time domains respectively. k and R k The diagonal matrix of .

[0037] When the control input u of the i-th leg is obtained by solving the quadratic programming problem i After that, the joint torque can be calculated through the following mapping relationship:

[0038]

[0039] Among them, the right leg joint torque τ1=[τ 11 τ 12 τ 13 τ 14 τ 15 τ 16 ] T , where: τ 11 represents the torque of the three hip joints around the x-axis, τ 12 represents the torque of the three hip joints around the y-axis, τ 13 represents the torque of the three hip joints around the z axis, τ 14 represents the torque of the knee joint around the y-axis, τ 15 represents the torque of the ankle joints around the y-axis, τ 16 represents the torque of the ankle joints around the x-axis, J i is the Jacobian matrix of the i-th leg, and its expression is:

[0040]

[0041] Among them, J v and J ω Denote the Jacobian matrix J i Linear velocity and angular velocity.

[0042] It should be noted that during the robot's motion, its feet are always in one of two states: support phase or swing phase, and the ground contact force can only act on the feet during the support phase. Therefore, the model predictive control algorithm is not suitable for controlling the swing leg. This system uses a Cartesian proportional-differential controller to achieve swing leg motion control, so that it can accurately track the desired foot end position. and speed The corresponding foot end force Fi can be expressed as:

[0043]

[0044] where k p and k d Represent the proportional control gain and the differential control gain respectively.

[0045] The gait generator determines the stance and swing phase states of the robot's foot within a fixed gait cycle. When the foot is in the stance phase, a model predictive control algorithm is used to implement force / position hybrid control of the stance leg. When the foot switches to the swing phase, a Cartesian PD controller is used to track the trajectory of the swing leg.

[0046] Step S4: Evaluation and Experimental Analysis: To verify the effectiveness of the proposed model predictive control algorithm, a humanoid bipedal robot simulation environment was built based on the MuJoCo physics engine. This platform, with its advanced dynamic modeling capabilities, provides high-fidelity verification of the control algorithm. During simulation, users trigger specific motion patterns by inputting control commands via the keyboard. These commands are injected into the system during the simulation initialization phase. Through a series of simulations and physical experiments, the control performance of the proposed MPC controller under various terrain conditions was systematically evaluated. Given the high consistency between the simulation results and the experimental data, the physical experimental data will be analyzed in particular. In the flat terrain walking test, the longitudinal velocity tracking performance demonstrated that the actual velocity effectively tracked the desired velocity. The output states of the robot's center of mass position and velocity did not exhibit significant abrupt changes. Upper body vibration was kept within a small range, and the actual output torque of each joint was less than the rated torque of the drive motor. The flat terrain walking experiments demonstrated that the proposed MPC controller effectively maintained the robot's walking balance and ensured gait stability. In the rough terrain walking test, the MPC controller also effectively tracked the target velocity command. Comparison and analysis of the results from the flat terrain walking experiment revealed that the robot was significantly affected by the terrain's undulations around the 15-second mark, but the MPC controller was able to maintain stability. While the required torques for each joint increased compared to flat terrain, none exceeded the rated torque of the drive motors. The experimental data fully demonstrated that the MPC controller ensured robust and stable walking on rough terrain despite external disturbances.

[0047] In summary, due to the adoption of the above technical solution, the beneficial effects of the present invention are:

[0048] 1. The present invention designs a bipedal humanoid robot with 12 degrees of freedom, and equips each leg with 6 joint drivers at the three main parts of the hip joint, knee joint and ankle joint, providing sufficient hardware conditions for the flexible movement of the self-developed robot.

[0049] 2. This paper simplifies the humanoid robot model using a single rigid-body model and designs a motion control method and constraints for the robot with 12 degrees of freedom. Although this simplified model ignores leg dynamics, it still achieves robust stability for a 12-degree-of-freedom system under multi-body effects, improving the system's adaptability and real-time performance.

[0050] 3. The present invention conducts motion control experiments on self-developed robots through simulation and actual objects, effectively verifying the effectiveness of the proposed model predictive control algorithm. BRIEF DESCRIPTION OF THE DRAWINGS

[0051] To facilitate understanding by those skilled in the art, the present invention is further described below with reference to the accompanying drawings.

[0052] Figure 1This is a structural diagram of the 12-DOF bipedal humanoid robot of the present invention.

[0053] Figure 2 This is a comparison diagram of the simulation and real robots of the present invention.

[0054] Figure 3 This is a longitudinal speed tracking diagram of the flat terrain walking test of the present invention.

[0055] Figure 4 This is a diagram showing the change in the robot's center of mass position during the flat terrain walking test of the present invention.

[0056] Figure 5 This is a diagram of the robot speed change during the flat terrain walking test of the present invention.

[0057] Figure 6 This is a diagram showing the changes in the Euler angles of the robot body during the flat terrain walking test of the present invention.

[0058] Figure 7 This is the longitudinal speed tracking diagram of the rough terrain walking test of the present invention.

[0059] Figure 8 This is a diagram showing the change in the center of mass position of the robot during the rough terrain walking test of the present invention.

[0060] Figure 9 This is a diagram of the robot speed change during the rough terrain walking test of the present invention.

[0061] Figure 10 This is a diagram showing the changes in Euler angles of the robot body during the rough terrain walking test of the present invention. DETAILED DESCRIPTION

[0062] The following will clearly and completely describe the technical solutions of the present invention with reference to the embodiments. Obviously, the embodiments described are only some of the embodiments of the present invention, rather than all of them. The embodiments of the present invention and all other embodiments obtained by ordinary technicians in this field without making creative efforts are within the scope of protection of the present invention.

[0063] The present invention provides a gait control method for a 12-degree-of-freedom bipedal humanoid robot, which includes the following steps:

[0064] Step S1: Structural design of the self-developed robot; the self-developed humanoid robot's legs are designed with 12 degrees of freedom to achieve flexible motion control. Each leg of the robot is equipped with six joint drive mechanisms, and the joints are distributed in three main parts: the hip, knee, and ankle, forming a complex motion system. The hip joint consists of three joints: hip roll, yaw, and pitch, which are responsible for the robot's rolling, yaw, and tilting movements in space respectively; the knee joint is equipped with a pitch joint, which is mainly responsible for leg flexion and extension; the ankle joint contains two joints: roll and pitch, which are responsible for the flexible adjustment of the robot's foot to adapt to different terrains and postures. The overall structure of the robot consists of a body link, a hip roll link, a hip yaw link, a knee pitch link, an ankle roll link, and an ankle pitch link. These links support the robot's complex rotation and movement in three-dimensional space and can maintain balance and stability while supporting the ground with both legs.

[0065] Step S2: Model the self-developed bipedal humanoid robot. Because the full-body dynamics of the bipedal humanoid robot are highly nonlinear and complex, a single rigid-body model is used for simplification to ensure the real-time performance of the model predictive control controller. Although this simplified model ignores the leg dynamics, the model predictive control-based controller still achieves robust stability for the high-degree-of-freedom system under multibody effects. The single rigid-body model is represented in the world coordinate system as follows:

[0066]

[0067] Represents the position coordinates of the center of mass, p x is the position of the center of mass along the x-axis, p y is the position of the center of mass along the y-axis, p z is the position of the center of mass along the z axis, m is the mass of the robot, and They represent the ground contact force and torque of leg i (i=1,2) in the world coordinate system, i=1 corresponds to the right leg of the robot, i=2 corresponds to the left leg, and are the longitudinal force, lateral force and vertical force respectively. xi , F yi and F zi are longitudinal force, lateral force and vertical force respectively, vector g=[0 0 g] T , where g is the acceleration due to gravity, and are the robot angular velocity and moment of inertia in the world coordinate system, is the distance vector from the robot's center of mass to the foot contact point.

[0068] Robot state variables Ground contact force and moment input u = [F1 F2 M1 M2] T , then the robot dynamics can be expressed in the following state space form:

[0069]

[0070] in

[0071]

[0072] Among them, σ1 and σ2 are Boolean variables used to indicate whether the foot end is in contact with the ground; I3 represents the 3×3 unit matrix; the skew-symmetric matrix Characterized by r i ×F i The cross product operation relationship of .

[0073] In order to eliminate the nonlinear term w in the robot dynamic state space equation, the robot state x s Corrected to new state variables Therefore, the state space equation can be rewritten in standard form as follows:

[0074]

[0075] in

[0076]

[0077] Step S3: Design a self-developed bipedal humanoid robot controller. To implement an MPC controller based on a linearized state-space dynamics model, the forward Euler method is used to discretize the robot's dynamics state-space equation at the kth time step, yielding:

[0078] x(k+1)=A k x(k)+B k u(k);

[0079] A k =I 13 +AT s , B k =BT s ;

[0080] Based on the discrete-time model, a model predictive control problem with a finite time domain h is constructed, and its quadratic cost function is expressed as follows:

[0081]

[0082] The constraints included are:

[0083] x(k+1)=A k x(k)+B k u(k)

[0084] -μF zi ≤F xi ≤μF zi

[0085] -μF zi ≤F yi ≤μF zi

[0086] F z，min ≤F zi ≤μF z，max

[0087] M x =0

[0088] -F z ·l t ≤M y ≤F z ·l h

[0089] F y ·l t -μ·M y -μ·l t ·F z ≤M z ≤μ·M y +μ·l t ·F z -F y ·l t

[0090] μ·l h ·F z -μ·M y -F y ·l h ≤M z ≤μ·M y +F y ·l h -μ·l h ·F z

[0091] The first term is used to track the desired state x r , the second one aims to minimize the control input to reduce the energy consumption of movement; Q k and R k is a diagonal weight matrix, which is used to adjust the dynamic response characteristics of the state variables and control inputs respectively. The control problem needs to meet the following constraints: (1) discrete dynamics constraint; (2) contact point friction constraint; (3) line foot dynamics constraint to ensure that the robot meets the contact torque stability criterion. Among the specific parameters, μ represents the ground friction coefficient, F z，min and F z，maxRepresent the minimum and maximum thresholds of vertical force, l h and l t They represent the distance from the foot contact point to the heel and toes respectively.

[0092] To ensure the real-time performance of model predictive control, the control problem represented by the quadratic cost function can be transformed into a quadratic programming problem. Specifically, the discrete-time dynamic model with a finite time domain h can be expressed as:

[0093] X=A q x(0)+B q U;

[0094] Where the state vector X=[x(1) T x(2) T … x(h) T ] T ;

[0095] Input vector U=[u(0) T u(1) T … u(h-1) T )] T , x(0) represents the initial state of the system, matrix A q and B q A containing h time domains in the future k and B k Matrix. From this, we can get the following quadratic programming problem:

[0096]

[0097] The constraints are:

[0098] A i U≤B i , A e U=B e ;

[0099] Among them, A i and B i is the inequality constraint matrix, A e and B e is the equality constraint matrix, and the matrices H and f can be expressed as:

[0100]

[0101] Among them, X r is a column vector containing the expected system state in the future h time domains, Q and R are the weight matrices Q containing the future h time domains respectively. k and R k The diagonal matrix of .

[0102] When the control input u of the i-th leg is obtained by solving the quadratic programming problem i After that, the joint torque can be calculated through the following mapping relationship:

[0103]

[0104] Among them, the right leg joint torque τ1=[τ 11 τ 12 τ 13 τ 14 τ 15 τ 16 ] T , where: τ 11 represents the torque of the three hip joints around the x-axis, τ 12 represents the torque of the three hip joints around the y-axis, τ 13 represents the torque of the three hip joints around the z axis, τ 14 represents the torque of the knee joint around the y-axis, τ 15 represents the torque of the ankle joints around the y-axis, τ 16 represents the torque of the ankle joints around the x-axis, J i is the Jacobian matrix of the i-th leg, and its expression is:

[0105]

[0106] Among them, J v and J ω Denote the Jacobian matrix J i Linear velocity and angular velocity.

[0107] It should be noted that during the robot's motion, its feet are always in one of two states: support phase or swing phase, and the ground contact force can only act on the feet during the support phase. Therefore, the model predictive control algorithm is not suitable for controlling the swing leg. This system uses a Cartesian proportional-differential controller to achieve swing leg motion control, so that it can accurately track the desired foot end position. and speed The corresponding foot end force Fi can be expressed as:

[0108]

[0109] where k p and k d Represent the proportional control gain and the differential control gain respectively.

[0110] The gait generator determines the stance and swing phase states of the robot's foot within a fixed gait cycle. When the foot is in the stance phase, a model predictive control algorithm is used to implement force / position hybrid control of the stance leg. When the foot switches to the swing phase, a Cartesian PD controller is used to track the trajectory of the swing leg.

[0111] Step S4: Evaluation and Experimental Analysis: To verify the effectiveness of the proposed model predictive control algorithm, a humanoid bipedal robot simulation environment was built based on the MuJoCo physics engine. This platform, with its advanced dynamic modeling capabilities, provides high-fidelity verification of the control algorithm. During simulation, users trigger specific motion patterns by inputting control commands via the keyboard. These commands are injected into the system during the simulation initialization phase. Through a series of simulations and physical experiments, the control performance of the proposed MPC controller under various terrain conditions was systematically evaluated. Given the high consistency between the simulation results and the experimental data, the physical experimental data will be analyzed in particular. In the flat terrain walking test, the longitudinal velocity tracking performance demonstrated that the actual velocity effectively tracked the desired velocity. The output states of the robot's center of mass position and velocity did not exhibit significant abrupt changes. Upper body vibration was kept within a small range, and the actual output torque of each joint was less than the rated torque of the drive motor. The flat terrain walking experiments demonstrated that the proposed MPC controller effectively maintained the robot's walking balance and ensured gait stability. In the rough terrain walking test, the MPC controller also effectively tracked the target velocity command. Comparison and analysis of the results from the flat terrain walking experiment revealed that the robot was significantly affected by the terrain's undulations around the 15-second mark, but the MPC controller was able to maintain stability. While the required torques for each joint increased compared to flat terrain, none exceeded the rated torque of the drive motors. The experimental data fully demonstrated that the MPC controller ensured robust and stable walking on rough terrain despite external disturbances.

[0112] The preferred embodiments of the present invention disclosed above are intended only to help illustrate the present invention. These preferred embodiments do not exhaustively describe all details, nor do they limit the present invention to specific embodiments. Obviously, many modifications and variations are possible based on the contents of this specification. These embodiments are selected and described in detail in this specification to better explain the principles and practical applications of the present invention, thereby enabling those skilled in the art to better understand and utilize the present invention. The present invention is limited only by the claims and their full scope and equivalents.

Claims

1. A gait control method for a 12-degree-of-freedom bipedal humanoid robot, characterized in that: The following steps are involved: Step S1: Structural design of the self-developed robot; the self-developed humanoid robot's legs are designed with 12 degrees of freedom to achieve flexible motion control. Each leg of the robot is equipped with six joint drive mechanisms, and the joints are distributed in three main parts: the hip, knee, and ankle, forming a complex motion system. The hip joint consists of three joints: hip roll, yaw, and pitch, which are responsible for the robot's rolling, yaw, and tilting movements in space respectively; the knee joint is equipped with a pitch joint, which is mainly responsible for leg flexion and extension; the ankle joint contains two joints: roll and pitch, which are responsible for the flexible adjustment of the robot's foot to adapt to different terrains and postures. The overall structure of the robot consists of a body link, a hip roll link, a hip yaw link, a knee pitch link, an ankle roll link, and an ankle pitch link. These links support the robot's complex rotation and movement in three-dimensional space and can maintain balance and stability while supporting the ground with both legs. Step S2: Model the self-developed bipedal humanoid robot. Because the full-body dynamics of the bipedal humanoid robot are highly nonlinear and complex, a single rigid-body model is used for simplification to ensure the real-time performance of the model predictive control controller. Although this simplified model ignores the leg dynamics, the model predictive control-based controller still achieves robust stability for the high-degree-of-freedom system under multibody effects. The single rigid-body model is represented in the world coordinate system as follows: Represents the position coordinates of the center of mass, p x is the position of the center of mass along the x-axis, p y is the position of the center of mass along the y-axis, p z is the position of the center of mass along the z axis, m is the mass of the robot, and They represent the ground contact force and torque of leg i (i=1, 2) in the world coordinate system, i=1 corresponds to the right leg of the robot, i=2 corresponds to the left leg, and are the longitudinal force, lateral force, and vertical force, respectively. xi , F yi , and F zi are longitudinal force, lateral force and vertical force respectively, vector g=[0 0 g] T , where g is the acceleration due to gravity, and are the robot angular velocity and moment of inertia in the world coordinate system, is the distance vector from the robot's center of mass to the foot contact point. Step S3: Design a self-developed bipedal humanoid robot controller. To implement an MPC controller based on a linearized state-space dynamics model, the forward Euler method is used to discretize the robot's dynamics state-space equation at the kth time step, yielding: x(k+1)=A k x(k)+B k u(k); in A k =I 13 +AT s ,B k =BT s ; Based on the discrete-time model, a model predictive control problem with a finite time domain h is constructed, and its quadratic cost function is expressed as follows: The constraints included are: x(k+1)=A k x(k)+B k u(k) -μF zi ≤F xi ≤μF zi -μF zi ≤F yi ≤μF zi F z，min ≤F zi ≤μF z，max M x ＝0 -F z ·l t ≤M y ≤F z ·l h F y ·l t -μ·M y -μ·l t ·F z ≤M z ≤μ·M y +μ·l t ·F z -F y ·l t μ·l h ·F z -μ·M y -F y ·l h ≤M z ≤μ·M y +F y ·l h -μ·l h ·F z The first term is used to track the desired state x r , the second one aims to minimize the control input to reduce the energy consumption of movement; Q k and R k is a diagonal weight matrix, which is used to adjust the dynamic response characteristics of the state variables and control inputs respectively. The control problem needs to meet the following constraints: (1) discrete dynamics constraint; (2) contact point friction constraint; (3) line foot dynamics constraint to ensure that the robot meets the contact torque stability criterion. Among the specific parameters, μ represents the ground friction coefficient, F z，min and F z，max Represent the minimum and maximum thresholds of vertical force, l h and l t Represents the distance from the foot contact point to the heel and toes respectively. To ensure the real-time performance of model predictive control, the control problem represented by the quadratic cost function can be transformed into a quadratic programming problem. Specifically, the discrete-time dynamic model with a finite time domain h can be expressed as: X=A q x(0)+B q U; Where the state vector X=[x(1) T x(2) T …x(h) T ] T ; Input vector U=[u(0) T u(1) T … u(h-1) T )] T , x(0) represents the initial state of the system, matrix A q and B q A containing h time domains in the future k and B k Matrix. From this, we can get the following quadratic programming problem: The constraints are: A i U≤B i ,A e U=B e ; Among them, A i and B i is the inequality constraint matrix, A e and B e is the equality constraint matrix, and the matrices H and f can be expressed as: Among them, X r is a column vector containing the expected system state in the future h time domains, Q and R are the weight matrices Q containing the future h time domains respectively. k and R k The diagonal matrix of . When the control input u of the i-th leg is obtained by solving the quadratic programming problem i After that, the joint torque can be calculated through the following mapping relationship: Among them, the right leg joint torque τ1=[τ 11 τ 12 τ 13 τ 14 τ 15 τ 16 ] T , where: τ 11 represents the torque of the three hip joints around the x-axis, τ 12 represents the torque of the three hip joints around the y-axis, τ 13 represents the torque of the three hip joints around the z axis, τ 14 represents the torque of the knee joint around the y-axis, τ 15 represents the torque of the ankle joints around the y-axis, τ 16 represents the torque of the ankle joints around the x-axis, J i is the Jacobian matrix of the i-th leg, and its expression is: Among them, J v and J ω Denote the Jacobian matrix J i Linear velocity and angular velocity. It should be noted that during the robot's motion, its feet are always in one of two states: support phase or swing phase, and the ground contact force can only act on the feet during the support phase. Therefore, the model predictive control algorithm is not suitable for controlling the swing leg. This system uses a Cartesian proportional-differential controller to achieve swing leg motion control, so that it can accurately track the desired foot end position. and speed The corresponding foot end force Fi can be expressed as: where k p and k d Represent the proportional control gain and the differential control gain respectively. The gait generator determines the stance and swing phase states of the robot's foot within a fixed gait cycle. When the foot is in the stance phase, a model predictive control algorithm is used to implement force / position hybrid control of the stance leg. When the foot switches to the swing phase, a Cartesian PD controller is used to track the trajectory of the swing leg. Step S4: Evaluation and Experimental Analysis: To verify the effectiveness of the proposed model predictive control algorithm, a humanoid bipedal robot simulation environment was built based on the MuJoCo physics engine. This platform, with its advanced dynamic modeling capabilities, provides high-fidelity verification of the control algorithm. During the simulation, the user triggers specific motion patterns by inputting control commands via the keyboard. These commands are injected into the system during the simulation initialization phase. Through a series of simulations and field experiments, the control performance of the proposed MPC controller under various terrain conditions was systematically evaluated. Given the high consistency between the simulation results and experimental data, the field experimental data will be analyzed in particular. During the flat terrain walking test, the longitudinal velocity tracking performance demonstrated that the actual velocity effectively tracked the desired velocity. The output states of the robot's center of mass position and velocity did not exhibit significant abrupt changes, upper body vibration was kept within a small range, and the actual output torque of each joint was less than the rated torque of the drive motor. The flat terrain walking test demonstrated that the proposed MPC controller effectively maintained the robot's walking balance and gait stability. During the rough terrain walking test, the MPC controller was able to effectively track the target velocity command. Comparison with the flat terrain walking test results revealed that the robot was significantly affected by the terrain fluctuations around 15 seconds, but the MPC controller was able to maintain stability. Although the required torques for each joint increased compared to the flat terrain walking test, none exceeded the rated torque of the drive motor. Experimental data fully proves that the MPC controller can ensure the robot's robust and stable walking on rough terrain under external interference conditions.

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