Constraint forces of the kinematic pair are the basis of the kinematics and dynamics analysis of mechanisms. Exploring the solution method for constraint forces is a hot issue in the mechanism theory fields. Based on the observation method and the theory of reciprocal screw system, the solution method of reciprocal screw system is improved and its solution procedures become easier. This method is also applied to the solution procedure of the constraint force. The specific expressions of the constraint force are represented by the reciprocal screw system of twist. The transformation formula of twist under different coordinates is given and it make the expression of the twist of kinematic pair more facility. A slider-crank mechanism and a single loop spatial RUSR mechanism are taken as examples. It confirms that this method can be used to solve the constraint force of the planar and spatial mechanism.

The constraint force analysis of kinematic pair is not only the key of using mechanisms reasonably and creating new mechanisms but also the important factor of kinematics and dynamics analysis and the base of structure design of mechanism. Traditional methods of constraint force analysis of kinematic pair include the graphic method and analytical method [

The reciprocal screw system represents the constraints and constraint forces acting on kinematic pairs. Since its solution method is complex and lacks commonality, based on Huang et al. [

A line vector (Figure

A line vector.

If

If

In order to decide the position of the axis of a screw,

The axis of a screw.

Since

The screw can be used to describe motions and forces. They are, respectively, called twist and wrench. The instantaneous twist of a rigid body can be written as

According to formula (

Therefore,

Generally, all the spatial forces acting on a rigid body can be reduced to a force

According to formula (

Therefore,

The specific Plücker coordinate of twist and wrench can be written as [

If

If

In order to calculate the reciprocal screw, a screw can be expressed by a row vector:

A kinematic pair is a combination of two kinematic bodies which have relative motion with respect to each other. Its motion can be described by screw system. The order of screw system is the same as the mobility. If the mobility of a kinematic pair is

Let

There are

The solution of the

If there are no zero rows in the matrix

where

where

The rest of elements of the

If there is a zero row vector in the matrix

where

where

The rest of elements of the

If

The reciprocal screw system of twist system is expressed as

The constraint wrench of the kinematic pair is written as

Expressions of twist of kinematic pair of spatial mechanism are hard to be obtained. It is necessary to set up a new coordinate frame. The rotation and displacement transformation matrix from the coordinate frames 2 to 1 are, respectively,

The twist of a kinematic pair in the coordinate frame 2 is written as

It can be divided into two parts:

The twist in the coordinate frame 1 is expressed as

It can be divided into two parts:

The twist in the coordinate frame 1 also can be expressed as

In the mechanism, wrenches acting on links can be classified into two types. One is the constraint wrenches of kinematic pair and the other is the external wrench. In Figure

The force figure of link

The common kinematic pairs in the planar mechanism mainly have the revolute pair and the prismatic pair. In the coordinate frame

The reciprocal screw system of the rotation pair of planar mechanism is given by

According to formulas (

The component force of the

According to formulas (

Similarly, the reciprocal screw system of the prismatic pair can be simplified as

According to formulas (

In order to verify the validity of the foregoing method, a slider-crank mechanism (Figure

The slider-crank mechanism.

The known geometry conditions are as follows:

Twists of kinematic pairs in the mechanism are written as

The reciprocal screw systems of kinematic pairs in the mechanism are given by

According to formulas (

Each wrench equilibrium equation can expand to 3 equations. Therefore, 9 equations can be obtained in total. Since there are 8 unknown solution coefficients and a trimming moment, the number of unknowns equals the number of equations. The analysis model can be solved. The constraint wrenches of kinematic pairs and the trimming moment of the slider-crank mechanism can be obtained:

The universal spatial kinematic pairs include the revolute pair, universal joint, spherical pair, and prismatic pair. In order to learn the solution method of the constraint wrenches in the spatial kinematic pairs, the single loop spatial RUSR mechanism is taken as example. Its schematic figure is shown in Figure

The single loop spatial RUSR mechanism.

The known conditions are as follows:

The coordinates of each kinematic pairs in the different coordinate frames are given by

In the following parts, notations are defined as

Twists of kinematic pairs in the different coordinate frames are shown as

The rotation transformation matrix from coordinate frames 2 to 1 are, respectively, shown as follows:

The displacement transformation matrix from coordinate frames 2 to 1 are, respectively, shown as

According to formula (

Based on the improved solution method of

According to formula (

According to formula (

Equations (

The constraint wrenches of kinematic pairs can be expressed by the reciprocal screw system. Based on this fact, the solution method of constraint wrenches is formed.

The improved solution can solve the reciprocal screw system by the programming conveniently and swiftly. It can enhance the solution efficiency of constraint wrenches.

The constraint wrench has an important significance for the analysis and application of the mechanism. According to the examples mentioned in this paper, the solution method of the constraint wrenches of kinematic pairs, which is based on the reciprocal screw, can solve the constraint wrenches of the planar and spatial mechanism.

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors gratefully acknowledge the financial support of the National Natural Science Foundation of China (Grant no. 51175422).