In order to promote the development of sports training robots and improve the training of lower limbs in Sanda, a new mechanical structure of Sanda training robots is designed. The thesis combines the mathematical method of fractional differential equations to design a new type of omnidirectional moving platform mechanism, which realises the movement in any direction in the plane and can turn at any radius during the movement. From the perspective of kinematics, the mathematical relationship between wheel speed and robot trajectory is analysed, and a virtual prototype technology combined with Pro/E and Adams joint modelling and simulation method is used to establish an accurate and reliable virtual prototype model. The experimental simulation results show that the designed virtual prototype model is consistent with the mathematical model, which verifies the practical feasibility of the mechanical structure of the lower limb power generation robot for Sanda movement, and provides a reliable basis for the establishment of the physical prototype.

#### Keywords

- Sanda exercise
- lower limb strength training
- lower limb strength training
- omnidirectional wheels
- virtual prototype technology

#### MSC 2010

- 12H99

In martial arts Sanda (hand), the side leg is often regarded as a ‘heavy weapon’ with high strength and lethality (heavily enough to stun the opponent on the spot and win the entire game). However, when completing the side leg movements, due to the large moment of inertia of the entire leg, the time required to start is relatively long. In addition, the trajectory (route) travelled is long, so the concealment of the movement is relatively poor and is easy to be seen by opponents, often in actual combat and short-sale or counter-attack. Therefore, in order to improve the actual combat effect of the side legs, how to increase the concealment and suddenness of the athlete's starting, how to increase the swing angular velocity and the striking speed and how to improve the effectiveness and pertinence of the striking should be the problems to be solved by training and scientific research [1,2,3].

This article mainly studies the structural design of the lower limb power robot with wheeled Sanda and its virtual prototype dynamics simulation and analyses whether the centre of gravity trajectory moves according to the expected trajectory under the known input to make the rationality of the mechanical structure design of the robot. Judge and give suggestions for optimisation.

The robot adopts omnidirectional wheel technology, and the omnidirectional wheel is composed of an active hub and a passive roller. The direction of the passive roller is parallel to the axis of the active hub. The movement of the wheel is a combination of the movement of the active hub and the passive roller. The four omnidirectional wheels are in a zigzag shape in the positive direction of the car body, and the omnidirectional wheels are 45° or 135° in the forward direction. Each omnidirectional wheel is independently driven by a servo motor. The speed of the four motors is controlled by a program, and then, the robot moves according to the combination of speeds. The robot motion structure is shown in Figure 1 [4, 5].

The numerical solutions to the initial value problems of ordinary differential equations are given in a recursive form, that is, the recursive algorithm. According to the recursive algorithm, when calculating _{k}_{+1}, the approximate value _{i}_{k}_{+1}, only the ‘information’ _{k}_{k}_{k}_{+1}) of the _{k}_{+1} system at a certain time in the future. Such a numerical solution is called the single-step method for the state _{k}_{+1}) of a dynamic process _{k}_{+1} I, not only the information _{k}

Based on the above considerations, the method given by appropriately taking the information _{i}_{k}_{k}_{+1}] with a linear combination of _{i}_{i}

In the improved Euler method,
^{3}) and the accuracy is higher than the local truncation error ^{2}) of the Euler formula, the following prediction-correction method is obtained:
_{−1} by itself.

In the above discussion, we have only considered the improvement of prediction methods, thinking from a methodological point of view, we should also consider the improvement of correction, and we can hope that the appropriate improvement will bring the return of increased accuracy. The following recommended solutions are commonly used [7]:
_{k}_{+1}, and the number of corrections

Similarly, available

Estimated with errors (8)–(11)

In integral formula,
_{n}_{i}_{i}_{i}_{i}

In particular, by replacing the integrand _{3}(_{i}_{i}_{i}_{i}

The four-node (_{i}_{i}_{i}_{i}_{3} (

Adams Forecast-Correction Formula:

Among them, _{n}_{n}_{n}

Adams software has strong analysis capabilities, but Adams’ modelling capabilities are generally not suitable for the construction of complex models; Pro/E has powerful modelling capabilities, which can build a variety of complex models, but Pro/E cannot perform complex kinematics and dynamics simulations. Combining the advantages of both, this paper uses Pro/E to model and then performs simulation analysis in Adams. The combined model of Pro/E and Adams is a process of continuous improvement. The 3D solid modelling includes two phases of part modelling and general assembly. According to the structural characteristics and functional requirements of the training robot, the modelling and assembly of each part are completed, and a three-dimensional solid model of the training robot is obtained. Through the interface program Mechanic/Pro between Pro/E and Adams software, rigid body and partial constraints are defined, and then, the 3D solid assembly model created by Pro/E is converted into Adams, which further improves the Adams dynamics model. The steps are shown in Figure 2 [9].

Pro/E was used to model the mechanical structure of the new Sanda lower limb powering robot. Figure 3 shows the parts model of the training robot, and Figure 4 shows the assembly model in Pro/E.

The Pro/E model is imported into Adams, and the Adams model is generated through interface software of mechpro2005 of Adams and Pro/E [10], and the simulation analysis is performed in Adams. Model transformation includes three steps: defining rigid body, defining constraints and model transformation. Add necessary constraints and forces to the system in the Adams environment and improve the imported Adams model. The necessary forces and drives are added to define the reference quantities of the material density, stiffness, elastic deformation and acceleration of gravity of each rigid body. Finally, the interaction forces between various rigid bodies are defined, including the contact force between the hub and the ground, the acceleration of gravity and the motion drive. The built Adams dynamic model is shown in Figure 5.

The specific statistics of each component, rigid body and constraint involved in the modelling process are shown in Table 1.

Modelling quantity statistics

Components | 42 | 1 main frame, 4 motor components, 4 main wheels, 32 hubs, 1 ground |

Defining rigid bodies | 43 | 1 main frame, 4 motor components, 4 main wheels, 32 auxiliary wheels, 1 ground, 1 rigid body by default |

Constraint | 37 | 36 rotating pairs, 1 locking pair |

Contact force | 32 | Is the force between the auxiliary wheels (32) and the ground |

Motion driver | 4 | Main wheels (4) rotation drive |

Gravitational acceleration | 1 | y-direction acceleration |

Given an omnidirectional wheel angular velocity of

Given the angular velocity of the omnidirectional wheel as

Figure 9 reflects the disturbance in the x-axis direction during the starting process. The maximum peak-to-peak value of the disturbance is 4 mm, and it becomes gentle after 15 s. According to the analysis, the cause of the disturbance is that the centre of gravity of the robot and its geometric centre are not in the same vertical direction, and the forces on the wheels are uneven when starting; it reflects the 1 mm disturbance on the y-axis change; the z-axis travels at a constant speed. Since the disturbance peaks sampled in Figure 9 are relatively small for the entire system, they have little effect on the overall system effect. Now of impact, the three velocity components of the hip and knee joints are relatively small, all <2 m/s. Among the X, Y and Z speed components of the ankle joint point, the speed in the Y direction is the smallest when it hits 4.03 m/s, while the speed component values in the X and Z directions have no significant difference (P> 0.05). The average is about 5.4 m/s. At the toes, there was no significant difference between the three speeds of X, Y and Z (P> 0.05), as now of on impact, with an average of about 7 m/s.

According to the dynamic simulation results obtained in Adams, it can be analysed that the centre of gravity of the robot is not at its geometric centre, which will certainly cause some interference to the control of the robot. Improving the design structure to make it more symmetrical can reduce this tendency of interference. From the perspective of the dynamic mathematical model analysis, Adams simulation results are consistent with the dynamic mathematical model. When the given input is the same, the dynamic simulation of the design is consistent with the mathematical model results, indicating that the design structure meets the training requirements.

The correspondence between wheel speed and trajectory was derived from the perspective of mathematical modelling. The feasibility of designing a lower limb power robot for wheeled Sanda is analysed by the method of joint modelling of Pro/E and Adams, and the consistency between the structural design and the mathematical model is verified. Adams kinematic analysis shows that the design meets the training requirements of lower limb strength training in Sanda, and a corresponding improvement plan is proposed for the shortcomings in the design. The vibration of the system is reduced by adding cushioning structures such as cushions; the symmetry of the system is improved to eliminate the trajectory shift caused by uneven force between the wheels.

#### Modelling quantity statistics

Components | 42 | 1 main frame, 4 motor components, 4 main wheels, 32 hubs, 1 ground |

Defining rigid bodies | 43 | 1 main frame, 4 motor components, 4 main wheels, 32 auxiliary wheels, 1 ground, 1 rigid body by default |

Constraint | 37 | 36 rotating pairs, 1 locking pair |

Contact force | 32 | Is the force between the auxiliary wheels (32) and the ground |

Motion driver | 4 | Main wheels (4) rotation drive |

Gravitational acceleration | 1 | y-direction acceleration |

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