| The underactuated mechanical system refers to a class of highly nonlinear mechanical systems,whose number of independent control variables is less than the number of degrees of freedom.As a branch of mechanical systems,it widely exists in the fields of transportation,engineering machinery,robots,and so on.Therefore,the research on the control theory of underactuated mechanical systems has always been a hot topic in the academic field.Although a large number of control methods and theories have been proposed for underactuated mechanical systems,there are still the following shortcomings:(1)When the current control methods deal with the second-order nonholonomic passive constraints of the underactuated mechanical system,one needs to transform the system dynamics equations or construct additional nonlinear partial differential equations,which makes the controller design more difficult.Also,it limits the application scope of the method and leads to increasing computational efforts in engineering applications.(2)The uncertainty model of the underactuated mechanical systems used at present cannot accurately reflect the essence of uncertainty,which leads to some application problems?for example,it makes the controlled underactuated mechanical system unable to work in the optimal state or not suitable for the actual underactuated mechanical system.(3)The selection of control parameters in current control methods is usually based on expert experience and the completion of control tasks as the standard,while ignoring the balance between system transient performance,steady-state performance,and control cost.In this dissertation,based on the Moore-Penrose generalized inverse,adaptive robust control theory,fuzzy set theory,and Nash game theory,a control method for uncertain underactuated mechanical systems is proposed.The main contents of this dissertation are as follows:(1)The inverse dynamics of the underactuated mechanical system is studied,and the analytical expression of a constraint-following force/torque is obtained.Firstly,based on the idea of constraint-following and the dynamic equation of the nominal underactuated mechanical system,the dynamic model of the nominal underactuated mechanical system subject to servo constraint is established,which makes the servo constraint equation and system acceleration(system dynamics)consistent in mathematical form.Secondly,based on the analysis of dynamics,the internal relationship between the servo constraint and the dynamic equation of the nominal underactuated mechanical system is revealed,and based on Moore-Penrose generalized inverse,the method of solving servo constraint force/torque of the nominal underactuated mechanical system is proposed.Then,based on the relationship between the servo constraint force/torque and the constraint-following control force/ torque in the range space and zero space,the analytical expression of the constraint-following control force/torque is derived.Finally,a numerical simulation based on a two-wheeled non-coaxial mobile robot is carried out to verify the effectiveness of the constraint-following control force/torque.(2)The dynamic model of the fuzzy uncertain underactuated mechanical system is built,and a constraint-following controller is designed.Firstly,aiming at the problem that the previous uncertainty models cannot accurately reflect the uncertainty,the dynamics model of the fuzzy uncertain underactuated mechanical system is established by combining the system theory and fuzzy set theory,where the degree of occurrence of events is used to reflect the boundary information of system uncertainty.Next,the idea of constraint-following is introduced,and the dynamic model of the fuzzy uncertain underactuated mechanical system with servo constraint is constructed.Then,aiming at the shortcomings of current control methods in dealing with second-order nonholonomic passive constraints,an analytical adaptive robust constraint-following controller is designed based on the inverse dynamics and the research result of adaptive robust control theory.Finally,the practical stability analysis proves that the designed controller can make the controlled system reach uniform boundedness and uniform ultimate boundedness.A numerical simulation based on a two-wheeled non-coaxial mobile robot is carried out to verify the effectiveness of the designed controller.(3)The problem formulation of the control parameter optimization problem is constructed,and subsequently,this optimization is solved and the optimal control parameters are obtained.Firstly,the system performance of the controlled underactuated mechanical system is analyzed,and the relationship between the system transient performance,steady-state performance,control cost,and dual control parameters are obtained.Secondly,the Nash game theory is introduced.Based on the internal relationship between the dual-parameter optimization problem and the Nash game decision problem,the performance index of the parameters is designed and the dual-parameter optimization problem is constructed.Then,the method of solving the dual-parameter optimization problem is proposed,and the Nash equilibrium value is obtained so that any parameter changing its value in this Nash equilibrium value(the other parameter value remains unchanged)will not reduce the value of its performance index.Finally,a numerical simulation based on a two-wheeled non-coaxial mobile robot is carried out to verify the effectiveness of the designed method.(4)Experiments are carried out on a two-wheeled non-coaxial mobile robot to verify the effectiveness and advantages of the proposed control method for uncertain underactuated mechanical systems in practical engineering problems.Firstly,based on the working principle of flywheel balance mode and handlebar balance mode of a two-wheel noncoaxial mobile robot,the robot test platform is built,including a mechanical system and a control system.Then,based on the flywheel balancing mode of the two-wheel non-coaxial mobile robot,the effectiveness of the proposed method is verified.The comparative test with the interconnection and damping assignment passivity-based control method verifies the advantages of the proposed method.Finally,the effectiveness of the proposed method is verified based on the handlebar balance mode of the two-wheel non-coaxial mobile robot,and the advantage of the proposed method is verified by the comparison test with the interconnection and damping assignment passivity-based control method. |
