Nonholonomic constrained robot

Holonomic and nonholonomic constraints

Nonholonomic constraint robot is mainly characterized by robot system dynamics satisfying nonholonomic constraint . For example, space robots , Mobile robot, etc . For any electromechanical system , If there are any nonholonomic constraints in the system , Then the Mechatronics System / A robot is a nonholonomic system . Physical systems can be classified into complete systems and non complete systems . One of the conditions of many theories or equations , All constraints in the system must be complete . for example , Suppose a physical system is a complete system and a unicast system , Then the necessary and sufficient condition for the establishment of Lagrange equation is Hamilton principle .

Unlike complete constraints , Nonholonomic constraints have special definitions . For complete restraint systems , The system can be represented by the following equation ：

f(x_1,x_2,......,x_n,t)=0

A nonholonomic system cannot be uniquely represented by the above formula . Complete constraint equations and positions 、 It's about time , It has nothing to do with speed . The main characteristic of nonholonomic systems is the existence of non integrable differential constraints . Nonholonomic systems exist in at least three conditions ： The object is rolling ; The constraints of the system include inequalities ; The constraints of the system are related to speed （ For example, the Pfaff constraint ）. in other words , If a system is expressed in the following form ：

f(x_1,x_2,......,x_n,\dot x_1,\dot x_2,......,\dot x_m,t)=0

If in the equation ,\dot x_1,\dot x_2,......,\dot x_m Non integrable , The system is incomplete .

An example that is very relevant to our life is “falling cat problem”, Falling cat problem In any initial position , When landing, they always land with their feet facing down , And the process does not violate the law of conservation of angular momentum . The explanation of the nature of the cat falling problem is more complicated , But it is a typical application of nonholonomic constraints .

Falling cat problem

The other is mobile robot , The wheels of the mobile trolley and the ground are rolling , The system has nonholonomic constraints

Mobile trolley

Free floating space robot , Conservation of angular momentum of the system , And the conservation equation of angular momentum can not be integrated to obtain the position level equation , Therefore, the system has nonholonomic constraints .

Free floating robot

All in all , If the robot constraint equation contains the derivative of coordinates to time ( Such as motion constraints ), And it is impossible to integrate the equation into a finite form , Such constraints are called nonholonomic constraints . Nonholonomic constraint equations are always in the form of differential equations . conversely , If the robot constraint equation does not include the derivative of coordinates to time , Or the differential term in the constraint equation can be integrated into a finite form , Such constraints are called holonomic constraints .

For nonholonomic constrained robot systems , Nonholonomic constraints are constraints on the derivatives of the generalized coordinates of the system , And it is generally a speed level constraint , Differential constraint . Nonholonomic constraints do not reduce the configuration degrees of freedom of the system . This makes the number of independent controls of the system less than the configuration degrees of freedom of the system , It brings great difficulties to its control design . in addition , The differential geometry method using the theory of nonlinear control system has been proved : Nonholonomic systems cannot be stabilized by continuous state feedback . Therefore, a large number of mature results in modern control theory, which mainly studies continuous state feedback, can not be directly applied to the stabilization control of nonholonomic systems , The research of nonholonomic control system has become one of the most challenging problems in the field of control .