# DOF Calculation Method of Planar Mechanisms Based on Constraint Normal Line

Authors: Guojun Luo; Xiaohui Wang; Xianguo Yan
Journal: Mathematical Problems in Engineering
 (2023)

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## Abstract

In the analysis and calculation of the degree of freedom (DOF) of mechanisms, it is generally a complicated problem to judge the virtual constraint correctly. Under what conditions do virtual constraints exist, and how many virtual constraints are there? Understanding these problems can contribute to effectively tackling the difficulty of calculating the DOF. With planar mechanisms as a research object, the constraints among various components are simplified into point constraints, and the common normal at a constraint point is referred to as normal. According to whether a constraint point can move relative to the frame in its normal direction, normals are divided into different categories. Based on the geometric theorem for judging the DOF of a workpiece, a set of geometric theorems for judging the DOF of components and their nature, over-constraint, and quantity are established. After the correlation between virtual constraints and over-constraints is clarified, the total number of virtual constraints in a mechanism is calculated. The analysis and calculation of the DOF of several typical planar mechanisms demonstrate that the new method is logically rigorous, simple, and intuitive, and the DOF of a mechanism can be accurately calculated.


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## Body

## 1. Introduction

In a mechanism, two or more components are constraining a component simultaneously, and different forms of virtual constraints frequently appear on all kinds of mechanisms [1]. Virtual constraints can improve the strength, stiffness, and motion stability of a mechanism while realizing some of its special functions. Therefore, some components and kinematic pairs that are not related to the motion of a mechanism are generally added to the design of a mechanism to meet its different needs [2, 3]. However, the existence of virtual constraints impedes the analysis and calculation of the DOF of a mechanism.The calculation of the DOF of a mechanism is the basic issue of the analysis of a mechanism, and it is a crucial part of judging whether the machine structures can work normally [4]. The Grubler–Kutzbach formula (referred to as the G-K formula) has been used for a long time and is given as follows [5, 6]:(1)F=6−λn−∑i=1j6−λ−fi,where λ is a parameter of an independent ring, n is the number of moving links, m is the number of joints, and fi is the degree of freedom of each ith joint.After hundreds of years of development, G-K formula has many forms, and for the calculation of the degree of freedom of planar mechanism, now commonly used form is as follows:(2)F=3n−∑i=1mPi,where n is the number of moving links and Pi is the number of constraints imposed on the mechanism by the ith joint.However, many “counter-examples” have appeared in the development of a mechanism [7, 8]. As a result, this formula is no longer a “universal” formula. The main reason is that the effect of virtual constraints on components is ignored in the process of calculating the DOF of a mechanism [9]. To manage the problem that the G-K formula cannot grapple with virtual constraints, researchers from various countries have been exploring and researching new theories and methods for the calculation of the DOF of a mechanism [10–13].Based on the intuitive method, Sun [14] established a coordinate system, marked the possible independent movement of each component, determined the number of public constraints of a mechanism, and substituted it into the formula to calculate the DOF of a mechanism. Nevertheless, the method requires a rich DOF constraint recognition. The number of public constraints can be accurately identified by the experience, and the analysis process is relatively abstract.To calculate the DOF of a mechanism, Gogo [15] proposed the concepts of the dimension and intersection of the branch chain motion parameters based on the linear transformation theory. Nonetheless, it is the difficulty of this method to establish the linear change matrix and the intersection of the branch chain motion parameters.Guo and Yu [16] defined the concepts of real constrained higher pairs, imaginary constrained higher pairs, fully constrained lower pairs, semi-constrained lower pairs, and additional condition numbers. Based on the analysis theory of the kinematic joint constraint characteristics, the DOF of the planar mechanisms was calculated. However, the key to the calculation method was accurately obtaining the various parameters in the formula, especially the determination of the additional condition number. The difficulty of the analysis method is that the condition number that satisfies the motion in each mechanism is different and thus needs to be analyzed separately.Huang et al. [17, 18] analyzed the linear correlation between the branch-constrained spiral system and the reverse spiral system for the nonholonomic detachable planar linkage mechanism. Then, they obtained the number of common constraints and virtual constraints of components and substituted them into the modified G-K formula to calculate the DOF of a mechanism. Unfortunately, this method needs to be based on the spiral theory, and this method depends on the profound and difficult spiral theory.Martı´nez and Ravani [19] proposed a calculation formula for the DOF expressed by the dimension of branch chain displacement subgroups and the dimension of composite subgroups following the theory of Lie group/Lie subgroup algebra. However, complicated Jacobian matrices are prone to appear in the mapping transformation, which makes the analysis and calculation more difficult.Osgouie and Gard [20] proposed a matrix method based on the rank of the Jacobian of the mechanism, and its application is investigated. It is shown that the matrix method will definitely lead to a correct answer; however, it is lengthy and consumes more computational effort.Zhang et al. [21] established a method of removing the virtual constraints in which the lower pairs were replaced by higher pairs. For the virtual constraints of trajectory coincidence in the plane kinematic pair and without reducing the number of components, the lower pair was replaced by the plane higher pairs. However, for the kinematic pairs with multiple overlapping trajectory points in the mechanism, it does not explain which kinematic pairs should be replaced, which also makes the application of this method have certain limitations.Rajashekhar and Debasish [22] proposed a method similar to zebra crossing to determine the mobility from the zebra crossing diagram so as to quickly calculate the DOF of the mechanism. This algorithm takes into account the number of patches between the black patches, the number of joints attached to the fixed link, and the number of loops in the mechanism so that the degree of freedom of the mechanism with virtual constraints can be accurately calculated.The above methods have their advantages in calculating the DOF of different types of mechanisms. However, these methods require rich analytical experience and a deep mathematical theory foundation, which are difficult for ordinary engineering and technical personnel to grasp. Moreover, the DOF of a mechanism is still unable to be accurately analyzed and judged.Wang et al. [23–25] analyzed the DOF of the workpiece in the fixture in the early stage and simplified the positioning of the workpiece in the fixture with several contact points. According to the number of common normal lines at the constraint point and its geometric relationship, a geometric theorem for judging the DOF and over-constraint of the workpiece was established. This study makes the analysis principles and method of the DOF of the workpiece become more scientific and rigorous.In this paper, the method is extended to the calculation of the DOF of planar mechanisms, and a more intuitive and easy-to-understand geometric method is established. According to the number of constraint normal lines (CNLs) and geometric relations, the DOF of the component and its properties are judged, and a simple and effective new method is designed to calculate the DOF of a planar mechanism.


## 2. The Geometric Theorem for Judging the DOF of a Mechanism

### 2.1. The Geometric Theorem for Judging the DOF of a Rigid Body

An unconstrained rigid body has three DOFs in a plane and moves in two directions and rotates in the plane. When the rigid body is in contact with the reference object, the DOF relative to the reference object will be restricted, and the corresponding DOF will be lost. The contact between the rigid body and the reference object can be equivalent to multipoint positioning. Besides, the constraint on the rigid body at the positioning point can be expressed by a set of CNLs (normal line) geometric relations. It is stipulated in this study that the constraint normal is represented by a straight line with arrows, the starting point is the constraint point, and the direction points to the constrained component. If two or more normals are in the same plane, the plane is called the normal plane of these normals.Theorem 1.
Single-normal line geometry theorem.
The rigid body has no relative movement in the normal direction at the point of contact with the reference object, suggesting that the freedom of movement of the rigid body in the normal line direction is restricted.Proof.
As expressed in Figure1(a), the rigid body 1 and the reference object 2 are in contact at point O, and the rigid body cannot move in the normal line direction. Thus, the freedom of movement of the rigid body in the normal line F direction is restricted.Figure 1
The relationship between the DOF of the component and the normal lines. (a) Single normal line. (b) Two parallel normal lines. (c) Two intersecting normal lines. 1, Rigid body. 2, Reference object.
(a)(b)(c)Theorem 2.
The geometric theorem of two parallel normal lines:
If there are two parallel normal lines on a rigid body, the freedom of movement of the rigid body in the normal line direction and the rotational DOF of the normal plane are restricted.Proof.
In Figure1(b), there are two parallel normal lines F1 and F2 at points O1 and O2 on rigid body 1, respectively, and the instantaneous center of the velocity of the rigid body is at infinity. Therefore, the rigid body cannot rotate in the normal plane while the freedom of movement of the rigid body in the normal line direction is restricted.Theorem 3.
The geometric theorem of two intersecting normal lines:
If two normal lines on a rigid body intersect at a point, the angle between the normal lines can be any angle within the range of (0°, 90°], the intersection normal lines can be equivalent to any two noncollinear normal lines at the intersecting point in the normal plane, restricting the freedom of movement of the rigid body in any two directions in the normal plane.Proof.
In Figure1(c), the rigid body 1 is in contact with the reference object 2 at points O1 and O2, and the intersection point O of the two normals F1 and F2 must be the instantaneous center of the velocity of the rigid body because the instantaneous center is in any direction in the normal plane. Since the speed of is equal to 0, the two normals F1 and F2 can be equivalent to any two nonoverlapping normal lines in the normal plane at the intersection point O. For example, it can be equivalent to normal lines F3 and F4, which limit the freedom of movement of the rigid body in any two directions in the normal plane.


### 2.2. The Normals of Typical Kinematic Pairs and Their Number and Distribution

Each adjacent component in the mechanism is positioned by the kinematic pair and generates relative motion. The two basic forms of motion are movement and rotation. According to the geometric theorem of rigid body positioning, the constraint relationship between the two components will not change after the contacted surfaces of the two surfaces in relative motion are simplified into a few specific constraint points. Meanwhile, the number of constraint points is equal to the number of DOF restricted by this kinematic pair.In the planar mechanisms, the constraints generated by higher pairs and lower pairs can be simplified into CNLs, which can be expressed by normal lines with different geometric relationships. As presented in Figure2(a), higher pairs can be simplified into the normal line F of the common perpendicular direction of component 2 to component 1 at the contact point O, restricting the freedom of movement of the component in the normal line direction. In Figure 2(b), the sliding pair can be simplified to generate two parallel CNLs, F1 and F2 perpendicular to the X-axis direction at O1 and O2 that do not share the same point, respectively, so as to constrain its freedom of movement in the Y-direction. And the DOF of rotation in the plane. In Figure 2(c), the revolute pair can be simplified into the intersecting CNL generated by the two constraint points on the contact surface. Component 2 restricts the two DOF of movement of component 1. This demonstrates no DOF of movement in the plane, only the DOF of rotation around point O.Figure 2
The normal representation method of the typical motion pair. (a) Higher pair. (b) Sliding pair. (c) Revolute pair.
(a)(b)(c)


### 2.3. Types of Normal Lines and Their Judgment Methods

#### 2.3.1. Types of Normal Lines

The mechanism is based on the movement of the initial moving parts and is transmitted through each component and kinematic pairs step by step. The process can cause the normal line on the constraint point of kinematic pairs of the subsequent component to move. It is stipulated that if the normal line generated by kinematic pairs at the position of the constraint point cannot move relative to the frame, the normal line is called the static normal line (SNL); if the constraint point moves relative to the frame in the direction of its normal line, the normal line of this type is called the moving normal line (abbreviated as MNL). Blue and red represent SNLs and MNLs, respectively.


#### 2.3.2. Judgment Method of SNL and Dynamic Normal Line

Rule 1.
The components are connected through revolute pairs. If the rotation center does not move, the normal line generated by the constraint point is the SNL. Meanwhile, the normal line on the connection between the rotation center of the lower component and the rotation center of the upper component is the SNL, and the other directions are the dynamic normal line.
As expressed in Figure3(a), the three links are sections in one plane, the frame is connected to component 1 through a revolute pair, and component 1 can rotate around O1. Considering that O1 is a stationary point, the normal line F4 of the constraint point O2 of the lower component 2 in the O1O2 direction is a SNL, and the normal line F3 in other directions is a MNL. Besides, the rotation center O2 of component 2 is not stationary, and the rotation center O3 of the lower component 3 is displaced relative to the O1 point in any direction. Thus, the two normal lines F5 and F6 at the O3 point are MNLs.Figure 3
Judgment of SNL and dynamic normal line.
(a)(b)Rule 2.
If a component can only move along a certain direction, the normal line of the lower component in the direction perpendicular to the moving direction is a SNL, and the normal line in the direction is a dynamic normal line.
This means that the constraint points of the lower component move in that direction when a component moves in a certain direction. If they do not move in the normal line direction perpendicular to the moving direction, the corresponding normals are SNLs, while the constraint points of the lower component move in other directions. Hence, the corresponding normals are dynamic normal lines. As suggested in Figure3(b), frame 1 in the figure is connected with component 2 through the sliding pair; component 2 can only move in the X-direction; the normal lines F1 and F2 perpendicular to the axis direction are the SNLs; component 2 and component 3 are connected through the revolute pairs; and the constraint point O of the component 3 can only move in the X-direction. Therefore, the O point can move relative to the frame in the X-direction, and the corresponding normal line F3 is the dynamic normal line. The O point has no displacement along the Y-direction in the plane, and the corresponding normal line F4 is the SNL.


### 2.4. Judgment of Parallel Components and Their DOFs and Over-Constraints

Each component is connected by a kinematic pair to form a mechanism. The component is called a parallel component when two or more kinematic pairs simultaneously constrain the same component.


#### 2.4.1. Effect of SNL on Freedom and Over-Constraint of Components

Since the DOF of each component in the mechanism is relative to that of the frame, the frame in the mechanism is taken as a reference. The constraint normal on each component directly acting on the frame must be the constraint on its DOF generated by the SNL. The constraint on the DOF of the component is the same as that of the rigid body. If a certain DOF of the component is repeatedly constrained, this DOF of the component is called over-constrained.Theorem 4.
Effect of collinear SNL on over-constraint of components:
Ifn (n > 1) normal lines overlap with the component, the component moves in its normal line direction with the number of over-constraints of n-1.Proof.
As indicated in Figure4, component 1 generates two CNLs F1 and F2 at the constraint points O1 and O2, which are both the DOFs of component 2 in the Y-direction. There is a repeated constraint. Component 2 has an over-constraint.Figure 4
Static normal line collinear over-constraint.Theorem 5.
The influence of SNLs in a plane on over-constraints of components:
(1) If a component hasn (n > 2) parallel normal lines in the plane, the number of constraints on the component in the normal plane is n-2; (2) if there are n (n > 2) SNLs intersected at a point in the plane, the DOF of the component in the normal plane is over-constrained, and the number of over-constraints is n-2; and (3) if a component has n (n > 3) SNLs intersecting at 1 point in a plane, the component has (n-3) over-constraints.Proof.
As shown in Figure5(a), frame 2 acts on component 1 through five constraint points, and five parallel SNLs are generated at the five constraint points to act on component 1 together. According to Theorem 2, any two parallel SNLs limit the movement of component 1 along the normal line direction and the rotational DOF in the plane. However, the other three parallel SNLs have the same constraint effect on parallel component 1. Moreover, there are repeated constraints, and component 1 has three over-constraints.
As exhibited in Figure5(b), frame 2 is outside the circular component 1, and five SNLs intersect at the O point of the five constraint points. According to Theorem 3, any two intersecting SNLs can limit the freedom of movement of component 1 in any two directions in the plane. Hence, the two freedoms of movement in any direction in the plane are limited, and the three additional parallel SNLs have the same constraint effect. It is the over-constraint of component 1 in the plane.
As suggested in Figure5(c), component 1 is affected by component 2 at four constraint points, resulting in four SNLs that intersect at different points. Theorem 3 indicates that two intersecting normal lines can be equivalent to any two noncollinear normal lines in the normal plane at the intersection point, limiting the freedom of movement of the component in any two directions in the normal plane. In this way, any three normals intersecting at different points on component 1 can be equivalent to two parallel SNLs intersecting with another SNL. For example, regarding the SNLs F1, F2, and F3, the parallel SNLs F1 and F2 constrain the freedom of movement in the normal direction and the freedom of rotation in the plane, and another SNL F3 constrains another freedom of movement in the plane, according to Theorem 2. As a result, the three DOFs of component 1 in the plane have been completely constrained, and the extra SNL F4 is the over-constrained part of component 1.Figure 5
Over-constraint of plane static normal lines. (a) Parallel static normal lines. (b) Intersecting common point static normal lines. (c) Intersecting noncommon point static normal lines.
(a)(b)(c)


#### 2.4.2. Effect of MNL on the Freedom and Over-Constraint of Components

Each SNL acting on any component will form a constraint on the component. It would constrain a certain DOF of the component or cause an over-constraint on the component. However, some dynamic normals acting on a component generally do not form constraints on the component. The dynamic normal lines that do not form constraints on the component are considered invalid, and the dynamic normal lines that form constraints on the component are effective.(1) Judgment of the Effective MNL and Its Influence on the DOF of Components.Theorem 6.
Judgment of the effective MNL:
(1) The motion ofn DOF in a component leads to m (m > n) points of the next one (or more) component moving to produce m dynamic normal lines; (2) m MNLs are transmitted to the same parallel component, either directly or indirectly. The normal lines satisfying the above two conditions are called effective MNLs, producing (m-n) constraints on the parallel component.Proof.
Figure6(a) indicates that the geometric relationship in the mechanism is AB = CD, AB ∥ CD. Component 3 has the action of SNLs F1 and F2. According to Theorem 2, component 3 has only one freedom of movement perpendicular to the normal lines F1 and F2 in the plane. Based on this movement DOF, component 4 generates translational normal lines F3 and F4. Assuming that construction 3 is not fixed, the two parallel normal lines become two parallel SNLs. Theorem 2 implies the freedom of movement along the normal line direction and the freedom of rotation in the plane of constraint component 4. Under the action of component 3, the movement of component 3 can be decomposed into the normal line directions F3 and F4, and component 4 can move in the normal line direction. Therefore, the normal lines F3 and F4 of the horizontal action only restrict the rotational DOF of component 4 in the plane.
As revealed in Figure6(b), the two intersecting dynamic normal lines F3 and F4 on the target component 4 are derived from a moving DOF of component 1 along the Y-direction. Assuming that component 1 is stationary, the two intersecting dynamic normal lines become two intersecting SNLs, constraining the movement DOF of component 4 in the X and Y-directions. However, component 4 can move in the Y-direction under the action of component 1. Therefore, the two intersecting dynamic normal lines F3 and F4 only constrain the mobility degree of component 4 along the X-direction.Figure 6
Judgment of an effective moving normal line.
(a)(b)Theorem 7.
Over-constraint judgment of components by effective MNLs:
Ifn MNLs acting on the component are collinear, the target component has (n-1) over-constraints. If the motion of n (n > 2) normal lines of normal action acting on the target component begins directly or indirectly from the motion of a component, the target component is subjected to (n-2) over-constraints in its normal plane. Besides, the target component has (n-2) moving over-constraints if the motion of n (n > 2) normals acting on the target component is directly or indirectly derived from the motion of a component, and the perpendicular lines of each MNL intersect at the constraint point.
Similar to the determination of SNL over-constraint, the component will have over-constraints if the effective MNL repeats the DOF of the constraint of the component.
(2) Judgment of Invalid Dynamic Normal Lines. Many MNLs in the mechanism are invalid MNLs. Geometric theorems for their judgment are provided to correctly identify invalid MNLs.Theorem 8.
Judgment of invalid MNL:
If one (two) MNL acting on a parallel component is derived directly or indirectly from one (two) motion of the component, these MNLs are invalid.Proof.
In most mechanisms, a movement DOF of the component will only produce one MNL of the subordinate component. Hence, these MNLs are invalid.
In Figure7(a), the normal line F3 along the bar direction cannot move. It is the SNL and constrains the freedom of movement of component 2 in the normal line direction. Due to the rotation of component 1, the normal line F4 on revolute pairs at point B is transformed into a MNL and does not constrain the DOF of component 2 in the normal line direction. It is an invalid MNL.
It can be observed in Figure1(b) from the normal line transfer law 1 that is based on the rotation of a component at point E, the normal line F8 on component 3 and the normal line F6 on component 5 are MNLs, whereas the MNL F generated by a motion of component 4 is induced by the movement of the rotational pair center at point D. In the direction of MNLs F5, F6, and F8, the component can move and does not limit the freedom of the component. Then, they are invalid MNLs.
Figures7(a) and 7(b) are combined into a five-bar mechanism, and component 2 and component 5 are combined into component 2 (5), as shown in Figure 7(c). Component 2 (5) has two branches and is a parallel component. The previous analysis suggests that only the SNL F3 of the two branches limits the freedom of movement of component 2 (5), while the dynamic normal lines F4, F5, and F6 are invalid normal lines.Figure 7
Judgment of an invalid moving normal line.
(a)(b)(c)Theorem 9.
Normal line set theorem:
The restricted DOF of a component is the set of the restricted DOF of the component by the dynamic and SNLs.
The DOF constraints of parallel components have both SNLs and effective dynamic normal lines. The number, geometric relations, and combination forms of normal lines are various. According to the set theorem of normals, the properties and quantities of the constrained DOF of components can be judged, and then the properties of DOF of components and the properties and quantities of the over-constrained DOF are analyzed.


## 2.1. The Geometric Theorem for Judging the DOF of a Rigid Body

An unconstrained rigid body has three DOFs in a plane and moves in two directions and rotates in the plane. When the rigid body is in contact with the reference object, the DOF relative to the reference object will be restricted, and the corresponding DOF will be lost. The contact between the rigid body and the reference object can be equivalent to multipoint positioning. Besides, the constraint on the rigid body at the positioning point can be expressed by a set of CNLs (normal line) geometric relations. It is stipulated in this study that the constraint normal is represented by a straight line with arrows, the starting point is the constraint point, and the direction points to the constrained component. If two or more normals are in the same plane, the plane is called the normal plane of these normals.Theorem 1.
Single-normal line geometry theorem.
The rigid body has no relative movement in the normal direction at the point of contact with the reference object, suggesting that the freedom of movement of the rigid body in the normal line direction is restricted.Proof.
As expressed in Figure1(a), the rigid body 1 and the reference object 2 are in contact at point O, and the rigid body cannot move in the normal line direction. Thus, the freedom of movement of the rigid body in the normal line F direction is restricted.Figure 1
The relationship between the DOF of the component and the normal lines. (a) Single normal line. (b) Two parallel normal lines. (c) Two intersecting normal lines. 1, Rigid body. 2, Reference object.
(a)(b)(c)Theorem 2.
The geometric theorem of two parallel normal lines:
If there are two parallel normal lines on a rigid body, the freedom of movement of the rigid body in the normal line direction and the rotational DOF of the normal plane are restricted.Proof.
In Figure1(b), there are two parallel normal lines F1 and F2 at points O1 and O2 on rigid body 1, respectively, and the instantaneous center of the velocity of the rigid body is at infinity. Therefore, the rigid body cannot rotate in the normal plane while the freedom of movement of the rigid body in the normal line direction is restricted.Theorem 3.
The geometric theorem of two intersecting normal lines:
If two normal lines on a rigid body intersect at a point, the angle between the normal lines can be any angle within the range of (0°, 90°], the intersection normal lines can be equivalent to any two noncollinear normal lines at the intersecting point in the normal plane, restricting the freedom of movement of the rigid body in any two directions in the normal plane.Proof.
In Figure1(c), the rigid body 1 is in contact with the reference object 2 at points O1 and O2, and the intersection point O of the two normals F1 and F2 must be the instantaneous center of the velocity of the rigid body because the instantaneous center is in any direction in the normal plane. Since the speed of is equal to 0, the two normals F1 and F2 can be equivalent to any two nonoverlapping normal lines in the normal plane at the intersection point O. For example, it can be equivalent to normal lines F3 and F4, which limit the freedom of movement of the rigid body in any two directions in the normal plane.


## 2.2. The Normals of Typical Kinematic Pairs and Their Number and Distribution

Each adjacent component in the mechanism is positioned by the kinematic pair and generates relative motion. The two basic forms of motion are movement and rotation. According to the geometric theorem of rigid body positioning, the constraint relationship between the two components will not change after the contacted surfaces of the two surfaces in relative motion are simplified into a few specific constraint points. Meanwhile, the number of constraint points is equal to the number of DOF restricted by this kinematic pair.In the planar mechanisms, the constraints generated by higher pairs and lower pairs can be simplified into CNLs, which can be expressed by normal lines with different geometric relationships. As presented in Figure2(a), higher pairs can be simplified into the normal line F of the common perpendicular direction of component 2 to component 1 at the contact point O, restricting the freedom of movement of the component in the normal line direction. In Figure 2(b), the sliding pair can be simplified to generate two parallel CNLs, F1 and F2 perpendicular to the X-axis direction at O1 and O2 that do not share the same point, respectively, so as to constrain its freedom of movement in the Y-direction. And the DOF of rotation in the plane. In Figure 2(c), the revolute pair can be simplified into the intersecting CNL generated by the two constraint points on the contact surface. Component 2 restricts the two DOF of movement of component 1. This demonstrates no DOF of movement in the plane, only the DOF of rotation around point O.Figure 2
The normal representation method of the typical motion pair. (a) Higher pair. (b) Sliding pair. (c) Revolute pair.
(a)(b)(c)


## 2.3. Types of Normal Lines and Their Judgment Methods

### 2.3.1. Types of Normal Lines

The mechanism is based on the movement of the initial moving parts and is transmitted through each component and kinematic pairs step by step. The process can cause the normal line on the constraint point of kinematic pairs of the subsequent component to move. It is stipulated that if the normal line generated by kinematic pairs at the position of the constraint point cannot move relative to the frame, the normal line is called the static normal line (SNL); if the constraint point moves relative to the frame in the direction of its normal line, the normal line of this type is called the moving normal line (abbreviated as MNL). Blue and red represent SNLs and MNLs, respectively.


### 2.3.2. Judgment Method of SNL and Dynamic Normal Line

Rule 1.
The components are connected through revolute pairs. If the rotation center does not move, the normal line generated by the constraint point is the SNL. Meanwhile, the normal line on the connection between the rotation center of the lower component and the rotation center of the upper component is the SNL, and the other directions are the dynamic normal line.
As expressed in Figure3(a), the three links are sections in one plane, the frame is connected to component 1 through a revolute pair, and component 1 can rotate around O1. Considering that O1 is a stationary point, the normal line F4 of the constraint point O2 of the lower component 2 in the O1O2 direction is a SNL, and the normal line F3 in other directions is a MNL. Besides, the rotation center O2 of component 2 is not stationary, and the rotation center O3 of the lower component 3 is displaced relative to the O1 point in any direction. Thus, the two normal lines F5 and F6 at the O3 point are MNLs.Figure 3
Judgment of SNL and dynamic normal line.
(a)(b)Rule 2.
If a component can only move along a certain direction, the normal line of the lower component in the direction perpendicular to the moving direction is a SNL, and the normal line in the direction is a dynamic normal line.
This means that the constraint points of the lower component move in that direction when a component moves in a certain direction. If they do not move in the normal line direction perpendicular to the moving direction, the corresponding normals are SNLs, while the constraint points of the lower component move in other directions. Hence, the corresponding normals are dynamic normal lines. As suggested in Figure3(b), frame 1 in the figure is connected with component 2 through the sliding pair; component 2 can only move in the X-direction; the normal lines F1 and F2 perpendicular to the axis direction are the SNLs; component 2 and component 3 are connected through the revolute pairs; and the constraint point O of the component 3 can only move in the X-direction. Therefore, the O point can move relative to the frame in the X-direction, and the corresponding normal line F3 is the dynamic normal line. The O point has no displacement along the Y-direction in the plane, and the corresponding normal line F4 is the SNL.


## 2.3.1. Types of Normal Lines

The mechanism is based on the movement of the initial moving parts and is transmitted through each component and kinematic pairs step by step. The process can cause the normal line on the constraint point of kinematic pairs of the subsequent component to move. It is stipulated that if the normal line generated by kinematic pairs at the position of the constraint point cannot move relative to the frame, the normal line is called the static normal line (SNL); if the constraint point moves relative to the frame in the direction of its normal line, the normal line of this type is called the moving normal line (abbreviated as MNL). Blue and red represent SNLs and MNLs, respectively.


## 2.3.2. Judgment Method of SNL and Dynamic Normal Line

Rule 1.
The components are connected through revolute pairs. If the rotation center does not move, the normal line generated by the constraint point is the SNL. Meanwhile, the normal line on the connection between the rotation center of the lower component and the rotation center of the upper component is the SNL, and the other directions are the dynamic normal line.
As expressed in Figure3(a), the three links are sections in one plane, the frame is connected to component 1 through a revolute pair, and component 1 can rotate around O1. Considering that O1 is a stationary point, the normal line F4 of the constraint point O2 of the lower component 2 in the O1O2 direction is a SNL, and the normal line F3 in other directions is a MNL. Besides, the rotation center O2 of component 2 is not stationary, and the rotation center O3 of the lower component 3 is displaced relative to the O1 point in any direction. Thus, the two normal lines F5 and F6 at the O3 point are MNLs.Figure 3
Judgment of SNL and dynamic normal line.
(a)(b)Rule 2.
If a component can only move along a certain direction, the normal line of the lower component in the direction perpendicular to the moving direction is a SNL, and the normal line in the direction is a dynamic normal line.
This means that the constraint points of the lower component move in that direction when a component moves in a certain direction. If they do not move in the normal line direction perpendicular to the moving direction, the corresponding normals are SNLs, while the constraint points of the lower component move in other directions. Hence, the corresponding normals are dynamic normal lines. As suggested in Figure3(b), frame 1 in the figure is connected with component 2 through the sliding pair; component 2 can only move in the X-direction; the normal lines F1 and F2 perpendicular to the axis direction are the SNLs; component 2 and component 3 are connected through the revolute pairs; and the constraint point O of the component 3 can only move in the X-direction. Therefore, the O point can move relative to the frame in the X-direction, and the corresponding normal line F3 is the dynamic normal line. The O point has no displacement along the Y-direction in the plane, and the corresponding normal line F4 is the SNL.


## 2.4. Judgment of Parallel Components and Their DOFs and Over-Constraints

Each component is connected by a kinematic pair to form a mechanism. The component is called a parallel component when two or more kinematic pairs simultaneously constrain the same component.


### 2.4.1. Effect of SNL on Freedom and Over-Constraint of Components

Since the DOF of each component in the mechanism is relative to that of the frame, the frame in the mechanism is taken as a reference. The constraint normal on each component directly acting on the frame must be the constraint on its DOF generated by the SNL. The constraint on the DOF of the component is the same as that of the rigid body. If a certain DOF of the component is repeatedly constrained, this DOF of the component is called over-constrained.Theorem 4.
Effect of collinear SNL on over-constraint of components:
Ifn (n > 1) normal lines overlap with the component, the component moves in its normal line direction with the number of over-constraints of n-1.Proof.
As indicated in Figure4, component 1 generates two CNLs F1 and F2 at the constraint points O1 and O2, which are both the DOFs of component 2 in the Y-direction. There is a repeated constraint. Component 2 has an over-constraint.Figure 4
Static normal line collinear over-constraint.Theorem 5.
The influence of SNLs in a plane on over-constraints of components:
(1) If a component hasn (n > 2) parallel normal lines in the plane, the number of constraints on the component in the normal plane is n-2; (2) if there are n (n > 2) SNLs intersected at a point in the plane, the DOF of the component in the normal plane is over-constrained, and the number of over-constraints is n-2; and (3) if a component has n (n > 3) SNLs intersecting at 1 point in a plane, the component has (n-3) over-constraints.Proof.
As shown in Figure5(a), frame 2 acts on component 1 through five constraint points, and five parallel SNLs are generated at the five constraint points to act on component 1 together. According to Theorem 2, any two parallel SNLs limit the movement of component 1 along the normal line direction and the rotational DOF in the plane. However, the other three parallel SNLs have the same constraint effect on parallel component 1. Moreover, there are repeated constraints, and component 1 has three over-constraints.
As exhibited in Figure5(b), frame 2 is outside the circular component 1, and five SNLs intersect at the O point of the five constraint points. According to Theorem 3, any two intersecting SNLs can limit the freedom of movement of component 1 in any two directions in the plane. Hence, the two freedoms of movement in any direction in the plane are limited, and the three additional parallel SNLs have the same constraint effect. It is the over-constraint of component 1 in the plane.
As suggested in Figure5(c), component 1 is affected by component 2 at four constraint points, resulting in four SNLs that intersect at different points. Theorem 3 indicates that two intersecting normal lines can be equivalent to any two noncollinear normal lines in the normal plane at the intersection point, limiting the freedom of movement of the component in any two directions in the normal plane. In this way, any three normals intersecting at different points on component 1 can be equivalent to two parallel SNLs intersecting with another SNL. For example, regarding the SNLs F1, F2, and F3, the parallel SNLs F1 and F2 constrain the freedom of movement in the normal direction and the freedom of rotation in the plane, and another SNL F3 constrains another freedom of movement in the plane, according to Theorem 2. As a result, the three DOFs of component 1 in the plane have been completely constrained, and the extra SNL F4 is the over-constrained part of component 1.Figure 5
Over-constraint of plane static normal lines. (a) Parallel static normal lines. (b) Intersecting common point static normal lines. (c) Intersecting noncommon point static normal lines.
(a)(b)(c)


### 2.4.2. Effect of MNL on the Freedom and Over-Constraint of Components

Each SNL acting on any component will form a constraint on the component. It would constrain a certain DOF of the component or cause an over-constraint on the component. However, some dynamic normals acting on a component generally do not form constraints on the component. The dynamic normal lines that do not form constraints on the component are considered invalid, and the dynamic normal lines that form constraints on the component are effective.(1) Judgment of the Effective MNL and Its Influence on the DOF of Components.Theorem 6.
Judgment of the effective MNL:
(1) The motion ofn DOF in a component leads to m (m > n) points of the next one (or more) component moving to produce m dynamic normal lines; (2) m MNLs are transmitted to the same parallel component, either directly or indirectly. The normal lines satisfying the above two conditions are called effective MNLs, producing (m-n) constraints on the parallel component.Proof.
Figure6(a) indicates that the geometric relationship in the mechanism is AB = CD, AB ∥ CD. Component 3 has the action of SNLs F1 and F2. According to Theorem 2, component 3 has only one freedom of movement perpendicular to the normal lines F1 and F2 in the plane. Based on this movement DOF, component 4 generates translational normal lines F3 and F4. Assuming that construction 3 is not fixed, the two parallel normal lines become two parallel SNLs. Theorem 2 implies the freedom of movement along the normal line direction and the freedom of rotation in the plane of constraint component 4. Under the action of component 3, the movement of component 3 can be decomposed into the normal line directions F3 and F4, and component 4 can move in the normal line direction. Therefore, the normal lines F3 and F4 of the horizontal action only restrict the rotational DOF of component 4 in the plane.
As revealed in Figure6(b), the two intersecting dynamic normal lines F3 and F4 on the target component 4 are derived from a moving DOF of component 1 along the Y-direction. Assuming that component 1 is stationary, the two intersecting dynamic normal lines become two intersecting SNLs, constraining the movement DOF of component 4 in the X and Y-directions. However, component 4 can move in the Y-direction under the action of component 1. Therefore, the two intersecting dynamic normal lines F3 and F4 only constrain the mobility degree of component 4 along the X-direction.Figure 6
Judgment of an effective moving normal line.
(a)(b)Theorem 7.
Over-constraint judgment of components by effective MNLs:
Ifn MNLs acting on the component are collinear, the target component has (n-1) over-constraints. If the motion of n (n > 2) normal lines of normal action acting on the target component begins directly or indirectly from the motion of a component, the target component is subjected to (n-2) over-constraints in its normal plane. Besides, the target component has (n-2) moving over-constraints if the motion of n (n > 2) normals acting on the target component is directly or indirectly derived from the motion of a component, and the perpendicular lines of each MNL intersect at the constraint point.
Similar to the determination of SNL over-constraint, the component will have over-constraints if the effective MNL repeats the DOF of the constraint of the component.
(2) Judgment of Invalid Dynamic Normal Lines. Many MNLs in the mechanism are invalid MNLs. Geometric theorems for their judgment are provided to correctly identify invalid MNLs.Theorem 8.
Judgment of invalid MNL:
If one (two) MNL acting on a parallel component is derived directly or indirectly from one (two) motion of the component, these MNLs are invalid.Proof.
In most mechanisms, a movement DOF of the component will only produce one MNL of the subordinate component. Hence, these MNLs are invalid.
In Figure7(a), the normal line F3 along the bar direction cannot move. It is the SNL and constrains the freedom of movement of component 2 in the normal line direction. Due to the rotation of component 1, the normal line F4 on revolute pairs at point B is transformed into a MNL and does not constrain the DOF of component 2 in the normal line direction. It is an invalid MNL.
It can be observed in Figure1(b) from the normal line transfer law 1 that is based on the rotation of a component at point E, the normal line F8 on component 3 and the normal line F6 on component 5 are MNLs, whereas the MNL F generated by a motion of component 4 is induced by the movement of the rotational pair center at point D. In the direction of MNLs F5, F6, and F8, the component can move and does not limit the freedom of the component. Then, they are invalid MNLs.
Figures7(a) and 7(b) are combined into a five-bar mechanism, and component 2 and component 5 are combined into component 2 (5), as shown in Figure 7(c). Component 2 (5) has two branches and is a parallel component. The previous analysis suggests that only the SNL F3 of the two branches limits the freedom of movement of component 2 (5), while the dynamic normal lines F4, F5, and F6 are invalid normal lines.Figure 7
Judgment of an invalid moving normal line.
(a)(b)(c)Theorem 9.
Normal line set theorem:
The restricted DOF of a component is the set of the restricted DOF of the component by the dynamic and SNLs.
The DOF constraints of parallel components have both SNLs and effective dynamic normal lines. The number, geometric relations, and combination forms of normal lines are various. According to the set theorem of normals, the properties and quantities of the constrained DOF of components can be judged, and then the properties of DOF of components and the properties and quantities of the over-constrained DOF are analyzed.


## 2.4.1. Effect of SNL on Freedom and Over-Constraint of Components

Since the DOF of each component in the mechanism is relative to that of the frame, the frame in the mechanism is taken as a reference. The constraint normal on each component directly acting on the frame must be the constraint on its DOF generated by the SNL. The constraint on the DOF of the component is the same as that of the rigid body. If a certain DOF of the component is repeatedly constrained, this DOF of the component is called over-constrained.Theorem 4.
Effect of collinear SNL on over-constraint of components:
Ifn (n > 1) normal lines overlap with the component, the component moves in its normal line direction with the number of over-constraints of n-1.Proof.
As indicated in Figure4, component 1 generates two CNLs F1 and F2 at the constraint points O1 and O2, which are both the DOFs of component 2 in the Y-direction. There is a repeated constraint. Component 2 has an over-constraint.Figure 4
Static normal line collinear over-constraint.Theorem 5.
The influence of SNLs in a plane on over-constraints of components:
(1) If a component hasn (n > 2) parallel normal lines in the plane, the number of constraints on the component in the normal plane is n-2; (2) if there are n (n > 2) SNLs intersected at a point in the plane, the DOF of the component in the normal plane is over-constrained, and the number of over-constraints is n-2; and (3) if a component has n (n > 3) SNLs intersecting at 1 point in a plane, the component has (n-3) over-constraints.Proof.
As shown in Figure5(a), frame 2 acts on component 1 through five constraint points, and five parallel SNLs are generated at the five constraint points to act on component 1 together. According to Theorem 2, any two parallel SNLs limit the movement of component 1 along the normal line direction and the rotational DOF in the plane. However, the other three parallel SNLs have the same constraint effect on parallel component 1. Moreover, there are repeated constraints, and component 1 has three over-constraints.
As exhibited in Figure5(b), frame 2 is outside the circular component 1, and five SNLs intersect at the O point of the five constraint points. According to Theorem 3, any two intersecting SNLs can limit the freedom of movement of component 1 in any two directions in the plane. Hence, the two freedoms of movement in any direction in the plane are limited, and the three additional parallel SNLs have the same constraint effect. It is the over-constraint of component 1 in the plane.
As suggested in Figure5(c), component 1 is affected by component 2 at four constraint points, resulting in four SNLs that intersect at different points. Theorem 3 indicates that two intersecting normal lines can be equivalent to any two noncollinear normal lines in the normal plane at the intersection point, limiting the freedom of movement of the component in any two directions in the normal plane. In this way, any three normals intersecting at different points on component 1 can be equivalent to two parallel SNLs intersecting with another SNL. For example, regarding the SNLs F1, F2, and F3, the parallel SNLs F1 and F2 constrain the freedom of movement in the normal direction and the freedom of rotation in the plane, and another SNL F3 constrains another freedom of movement in the plane, according to Theorem 2. As a result, the three DOFs of component 1 in the plane have been completely constrained, and the extra SNL F4 is the over-constrained part of component 1.Figure 5
Over-constraint of plane static normal lines. (a) Parallel static normal lines. (b) Intersecting common point static normal lines. (c) Intersecting noncommon point static normal lines.
(a)(b)(c)


## 2.4.2. Effect of MNL on the Freedom and Over-Constraint of Components

Each SNL acting on any component will form a constraint on the component. It would constrain a certain DOF of the component or cause an over-constraint on the component. However, some dynamic normals acting on a component generally do not form constraints on the component. The dynamic normal lines that do not form constraints on the component are considered invalid, and the dynamic normal lines that form constraints on the component are effective.(1) Judgment of the Effective MNL and Its Influence on the DOF of Components.Theorem 6.
Judgment of the effective MNL:
(1) The motion ofn DOF in a component leads to m (m > n) points of the next one (or more) component moving to produce m dynamic normal lines; (2) m MNLs are transmitted to the same parallel component, either directly or indirectly. The normal lines satisfying the above two conditions are called effective MNLs, producing (m-n) constraints on the parallel component.Proof.
Figure6(a) indicates that the geometric relationship in the mechanism is AB = CD, AB ∥ CD. Component 3 has the action of SNLs F1 and F2. According to Theorem 2, component 3 has only one freedom of movement perpendicular to the normal lines F1 and F2 in the plane. Based on this movement DOF, component 4 generates translational normal lines F3 and F4. Assuming that construction 3 is not fixed, the two parallel normal lines become two parallel SNLs. Theorem 2 implies the freedom of movement along the normal line direction and the freedom of rotation in the plane of constraint component 4. Under the action of component 3, the movement of component 3 can be decomposed into the normal line directions F3 and F4, and component 4 can move in the normal line direction. Therefore, the normal lines F3 and F4 of the horizontal action only restrict the rotational DOF of component 4 in the plane.
As revealed in Figure6(b), the two intersecting dynamic normal lines F3 and F4 on the target component 4 are derived from a moving DOF of component 1 along the Y-direction. Assuming that component 1 is stationary, the two intersecting dynamic normal lines become two intersecting SNLs, constraining the movement DOF of component 4 in the X and Y-directions. However, component 4 can move in the Y-direction under the action of component 1. Therefore, the two intersecting dynamic normal lines F3 and F4 only constrain the mobility degree of component 4 along the X-direction.Figure 6
Judgment of an effective moving normal line.
(a)(b)Theorem 7.
Over-constraint judgment of components by effective MNLs:
Ifn MNLs acting on the component are collinear, the target component has (n-1) over-constraints. If the motion of n (n > 2) normal lines of normal action acting on the target component begins directly or indirectly from the motion of a component, the target component is subjected to (n-2) over-constraints in its normal plane. Besides, the target component has (n-2) moving over-constraints if the motion of n (n > 2) normals acting on the target component is directly or indirectly derived from the motion of a component, and the perpendicular lines of each MNL intersect at the constraint point.
Similar to the determination of SNL over-constraint, the component will have over-constraints if the effective MNL repeats the DOF of the constraint of the component.
(2) Judgment of Invalid Dynamic Normal Lines. Many MNLs in the mechanism are invalid MNLs. Geometric theorems for their judgment are provided to correctly identify invalid MNLs.Theorem 8.
Judgment of invalid MNL:
If one (two) MNL acting on a parallel component is derived directly or indirectly from one (two) motion of the component, these MNLs are invalid.Proof.
In most mechanisms, a movement DOF of the component will only produce one MNL of the subordinate component. Hence, these MNLs are invalid.
In Figure7(a), the normal line F3 along the bar direction cannot move. It is the SNL and constrains the freedom of movement of component 2 in the normal line direction. Due to the rotation of component 1, the normal line F4 on revolute pairs at point B is transformed into a MNL and does not constrain the DOF of component 2 in the normal line direction. It is an invalid MNL.
It can be observed in Figure1(b) from the normal line transfer law 1 that is based on the rotation of a component at point E, the normal line F8 on component 3 and the normal line F6 on component 5 are MNLs, whereas the MNL F generated by a motion of component 4 is induced by the movement of the rotational pair center at point D. In the direction of MNLs F5, F6, and F8, the component can move and does not limit the freedom of the component. Then, they are invalid MNLs.
Figures7(a) and 7(b) are combined into a five-bar mechanism, and component 2 and component 5 are combined into component 2 (5), as shown in Figure 7(c). Component 2 (5) has two branches and is a parallel component. The previous analysis suggests that only the SNL F3 of the two branches limits the freedom of movement of component 2 (5), while the dynamic normal lines F4, F5, and F6 are invalid normal lines.Figure 7
Judgment of an invalid moving normal line.
(a)(b)(c)Theorem 9.
Normal line set theorem:
The restricted DOF of a component is the set of the restricted DOF of the component by the dynamic and SNLs.
The DOF constraints of parallel components have both SNLs and effective dynamic normal lines. The number, geometric relations, and combination forms of normal lines are various. According to the set theorem of normals, the properties and quantities of the constrained DOF of components can be judged, and then the properties of DOF of components and the properties and quantities of the over-constrained DOF are analyzed.


## 3. Determination of Virtual Constraint

### 3.1. Sufficient and Necessary Conditions for the Formation of Virtual Constraints

The geometric theorem reveals that the constraint or over-constraint of the component is instantaneous. If in the process of a mechanism’s movement, a certain over-constraint can continuously and continuously constrain a certain DOF of the parallel component. Such a continuous over-constraint is called a virtual constraint. If the mechanism is over-constrained at a certain position while the over-constraint changes the constraint type after the mechanism moves slightly, making the over-constraint disappear in the component, such an over-constraint cannot become a virtual constraint. Therefore, the existence of over-constraints in parallel components is only a necessary condition for the existence of virtual constraints. Satisfying the over-constraints of speed matching is a sufficient and necessary condition for the existence of virtual constraints in parallel components.


### 3.2. The Method of Judging the Speed Matching of the Constrained Components

① When the SNLs at each restraint point of the restraint member intersect at one point, if the speed of rotation of each restraint point around the center of the velocity and the normal line can ensure that the angular velocity is equal, the velocity matches.Proof.
As shown in Figure8(a), components 1, 2, and 3 are rotated relative to component 4, and there are SNLs F1, F2, and F3 compared to Point O1, so component 4 is constrained in the plane. Since the mechanism satisfies the geometric relationship, CAD is a right angle, and Point B is the midpoint of CD, so that the angular velocity of each constrained point B, C, and D around Point O1 is always the same, satisfying the speed matching. However, in Figure 8(b), although there are also SNLs F1, F2, and F3 in B, C, and D compared to Point O1, the angle is not obtuse, and the geometric relationship cannot make the three SNLs intersect at point O1 all the time; it will become an intersecting SNLs with more than one intersection point in the plane. According to the normal geometry theorem, it can be determined that the DOF of component 4 is limited, and the CNLs are all real constraints.
② If the SNLs at each restraint point of the restraint member are parallel, if the linear velocity direction of each restraint point is the same and the size is equal, the speed matches.Figure 8
Constraint normal lines intersection. (a) Redundant constraint. (b) Over-constraint.
(a)(b)Proof.
Figure9 indicates that parallel component 4 has an over-constraint in both parallel bar mechanisms. In the process of mechanism movement, it can be ensured that the velocity components V1, V2, and V3 decomposed to parallel component 4 are the same, as the lengths of the three rods in Figure 9(a) are equal and the rotation angles of the three rods are equal. Moreover, the velocity matches. The three SNLs F1, F3, and F5 are always parallel. In other words, the over-constraint is continuous, and the over-constraint is virtual. In Figure 9(b), the lengths of the three bars are different, and the rotation angles of the three bars are different in the process of mechanism movement. Consequently, it is impossible to guarantee that the three SNLs F1, F3, and F5 are always parallel. If the mechanism changes one position, the three SNLs F1, F3, and F5 will not intersect in total. Thus, the three DOFs of parallel component 4 are restricted, and the mechanism cannot move. The over-constraint cannot constitute a virtual constraint.Figure 9
Constraint normal lines parallel. (a) Redundant constraint. (b) Over-constraint.
(a)(b)


### 3.3. Method and Steps of Redundant Restraint Judgment in the Mechanism

(1)
Based on the relationship between the various components in the mechanism, initially find the components that may be constrained, and then determine whether the components are constrained according to the constrained normal theorem. If the components are constrained, then calculate the number of constrained normal lines.(2)
Analyze and calculate the speed matching of the constrained components(3)
If the above two conditions are met, then determine whether the component is an instantaneous virtual constraint(4)
Determine whether the component has virtual constraints


## 3.1. Sufficient and Necessary Conditions for the Formation of Virtual Constraints

The geometric theorem reveals that the constraint or over-constraint of the component is instantaneous. If in the process of a mechanism’s movement, a certain over-constraint can continuously and continuously constrain a certain DOF of the parallel component. Such a continuous over-constraint is called a virtual constraint. If the mechanism is over-constrained at a certain position while the over-constraint changes the constraint type after the mechanism moves slightly, making the over-constraint disappear in the component, such an over-constraint cannot become a virtual constraint. Therefore, the existence of over-constraints in parallel components is only a necessary condition for the existence of virtual constraints. Satisfying the over-constraints of speed matching is a sufficient and necessary condition for the existence of virtual constraints in parallel components.


## 3.2. The Method of Judging the Speed Matching of the Constrained Components

① When the SNLs at each restraint point of the restraint member intersect at one point, if the speed of rotation of each restraint point around the center of the velocity and the normal line can ensure that the angular velocity is equal, the velocity matches.Proof.
As shown in Figure8(a), components 1, 2, and 3 are rotated relative to component 4, and there are SNLs F1, F2, and F3 compared to Point O1, so component 4 is constrained in the plane. Since the mechanism satisfies the geometric relationship, CAD is a right angle, and Point B is the midpoint of CD, so that the angular velocity of each constrained point B, C, and D around Point O1 is always the same, satisfying the speed matching. However, in Figure 8(b), although there are also SNLs F1, F2, and F3 in B, C, and D compared to Point O1, the angle is not obtuse, and the geometric relationship cannot make the three SNLs intersect at point O1 all the time; it will become an intersecting SNLs with more than one intersection point in the plane. According to the normal geometry theorem, it can be determined that the DOF of component 4 is limited, and the CNLs are all real constraints.
② If the SNLs at each restraint point of the restraint member are parallel, if the linear velocity direction of each restraint point is the same and the size is equal, the speed matches.Figure 8
Constraint normal lines intersection. (a) Redundant constraint. (b) Over-constraint.
(a)(b)Proof.
Figure9 indicates that parallel component 4 has an over-constraint in both parallel bar mechanisms. In the process of mechanism movement, it can be ensured that the velocity components V1, V2, and V3 decomposed to parallel component 4 are the same, as the lengths of the three rods in Figure 9(a) are equal and the rotation angles of the three rods are equal. Moreover, the velocity matches. The three SNLs F1, F3, and F5 are always parallel. In other words, the over-constraint is continuous, and the over-constraint is virtual. In Figure 9(b), the lengths of the three bars are different, and the rotation angles of the three bars are different in the process of mechanism movement. Consequently, it is impossible to guarantee that the three SNLs F1, F3, and F5 are always parallel. If the mechanism changes one position, the three SNLs F1, F3, and F5 will not intersect in total. Thus, the three DOFs of parallel component 4 are restricted, and the mechanism cannot move. The over-constraint cannot constitute a virtual constraint.Figure 9
Constraint normal lines parallel. (a) Redundant constraint. (b) Over-constraint.
(a)(b)


## 3.3. Method and Steps of Redundant Restraint Judgment in the Mechanism

(1)
Based on the relationship between the various components in the mechanism, initially find the components that may be constrained, and then determine whether the components are constrained according to the constrained normal theorem. If the components are constrained, then calculate the number of constrained normal lines.(2)
Analyze and calculate the speed matching of the constrained components(3)
If the above two conditions are met, then determine whether the component is an instantaneous virtual constraint(4)
Determine whether the component has virtual constraints


## 4. Establishment of the General Formula for the DOF of the Planar Mechanisms

If there aren movable components in a mechanism, the total number of DOFs is 3n when all components are in a free state. Assuming that there are m kinematic pairs in the mechanism and the number of DOFs restricted by each kinematic pair is pi, the total number of DOFs restricted by each kinematic pair is ∑i=1mPi. If there are a total of V virtual constraints in the mechanism (the V constraints do not play the role of restricting the DOF), the V constraints do not play the role of restricting the DOF, so the total number of the actual constraints of the DOF between the various components in the mechanism is ∑i=1mpi+V.The formula for calculating the freedom of the planar mechanisms is as follows:(3)F=3n−∑i=1mpi−V=3n−∑i=1mpi+V.In formula,F is the DOF of mechanism. N is the total number of moving components in the mechanism. m is the total number of kinematic pairs in the mechanism. pi is the number of DOF of constraints of each kinematic pair. V is the total number of virtual constraints in the mechanism.


## 5. Case Analysis

### 5.1. Analysis of the DOF of the Equal Diameter CAM Mechanism

Figure10 exhibits a simplified diagram of an equal-diameter CAM mechanism. Component 2 is the CAM, and the axis of the sliding pairs of component 3 is parallel at A and B. Component 2 generates collinear constraint normals F1 and F2 on component 3 at two points, O1 and O2, respectively. There are rotational DOFs in the plane based on the connection between component 2 and the frame through the revolute pairs at point O, implying that F1 and F2 are MNLs. According to Theorem 7, the two MNLs acting on component 3 are collinear. Hence, there is an over-constraint.Figure 10
The equal diameter cam mechanism.Based on the geometry relationship, the frame on both ends ofA and B by the prismatic pair connected to component 3 can be simplified into four parallel SNLs of F3, F4, F5, and F6 owing to the kinematic pairs of axial parallels. According to Theorem 5, any two parallel SNLs constrain the freedom of movement and rotation of component 3 in the X-direction, and the other two parallel SNLs are over-constraints.In the process of a mechanism’s movement, the positions and directions of parallel SNLsF3 and F4, F5, and F6 on the parallel component do not change, while the positions of dynamic normals F1 and F2 change but are always collinear. Hence, the over-constraints on the parallel component 3 are continuous. In this way, the three over-constraints on component 3 are virtual constraints, namely, V = 3.The mechanism has 2 movable components,n = 2, 2 higher pairs (each higher pair is limited to one DOF), 1 revolute pair (each revolute pair is limited to 2 DOFs), and 2 sliding pairs (each of which is limited to 2 DOFs, and the number of virtual constraints is V = 3). The DOFs can be obtained by substituting the following formula:(4)F=3n−∑i=1mpi+V=3×2−1×2−2×1+2+3=1.


### 5.2. Analysis of the DOF of the Press Mechanism

Figure11 presents a simplified diagram of the planar mechanisms of the press. The geometric conditions are that all components and the kinematic pairs of a mechanism are symmetrical along the line where component 1 is located in the plane. Component 8 is jointly restrained by the two rods 5 and 6 and the frame. Therefore, component 8 is a parallel component of a mechanism. Component 1 and the frame are connected by a sliding pair. According to the simplification of the plane kinematic pair normal line, component 1 has the function of parallel SNLs F1 and F2 in the X-direction. Theorem 2 suggests that the freedom of movement of component 1 in the X-direction and the rotation freedom in the plane are restricted, and only the freedom of movement is perpendicular to the normal line along the Y-direction.Figure 11
The press mechanism.One DOF of component 1 drives the movement of components 2 and 3 to produce two branches, which act on component 8 together to produce two intersecting MNLsF3 and F4. Theorem 6 reveals that the intersecting MNLs restrict the freedom of movement of the constrained component. Based on the initial movement of component 1 with downward freedom of movement, the intersecting effective MNL constrains the freedom of movement of component 8 in the X-direction. Besides, two parallel SNLs F5 and F6 are generated since the frame and component 8 are connected by a kinematic pair. The movement DOF and plane of component 8 in the X-direction are generated. The DOF of rotation within is restricted. The analysis of Theorem 9 normal line geometric theorem demonstrates that the effective intersecting dynamic normal line and the parallel SNL both restrict the freedom of movement of component 8 in the X-direction, and component 8 has an over-constraint.According to the geometric conditions, all components of a mechanism and the kinematic pair are symmetric along the axis of component 1 in the plane. As a result, the velocity components transmitted from component 5 and component 6 to component 8 in theY-direction are the same, implying velocity matching. In the process of movement, the geometric relationship of the normal lines on the parallel component does not change. Thus, the over-constraint has continuity and constitutes the virtual constraint: V = 1.The mechanism has 8 movable components,n = 8, 8 revolute pairs, and 4 sliding pairs. The number of virtual constraints is V = 1. The DOFs can be obtained by substituting the formula:(5)F=3n−∑i=1mpi+V=3×8−2×8+4+1=1.


### 5.3. DOF Analysis of the Planar 2-RRP Mechanism

The planar 2-RRP mechanism is exhibited in Figure12. The geometric conditions are CD∥EG and CE∥DG. According to the schematic diagram of a mechanism, component 2 has two branches. Thus, component 2 is the parallel component in the mechanism. Component 1 affects component 2 through rotation at point B. The judgment of the normal line type suggests that F3 is the freedom of movement of component 2 constrained by the SNL in the normal line direction, and F4 is an invalid dynamic normal line.Figure 12
The 2-RRP mechanism.The other branch acts on component 2 through components 4 and 3, and component 3 has the function of the SNLF7 to restrict the freedom of movement along rod 4. Additionally, there are two DOFs in the plane. On this basis, component 3 generates four effective horizontal action normal lines F5 and F6, F9 and F10 at points C and G in component 2 through the sliding pair. According to Theorem 6, the MNL has two over-constraints. Furthermore, the DOF of component 2 restricted based on component 3 is the same as the DOF of component 3. Therefore, component 2 is not constrained by the normal line of the flat behavior. The comprehensive analysis of Theorem 9 indicates that component 2 has two over-constraints.In the process of a mechanism’s movement, the effective MNLsF5 and F6, F9, and F10 are always parallel, and the constraints of the normal line of parallel component 2 do not change. Hence, the over-constraint on component 2 is the virtual constraint, and the number of virtual constraints of the whole mechanism is V = 2.The mechanism has 4 moving components,n = 4, 4 revolute pairs, and 2 kinematic pairs. The number of virtual constraints is V = 2. The DOFs can be obtained by substituting the following formula:(6)F=3n−∑i=1mpi+V=3×4−2×4+2+2=2.


### 5.4. DOF Analysis of the Lifting Shear Frame Mechanism

Figure13 shows a lifting shear frame that is hinged by multiple parallelograms and can obtain a larger telescopic stroke. It is a commonly used expandable telescopic mechanism for maintenance and warehouses. Component 1 and the rack are connected by a moving pair. It can be seen from Theorem 2 that two parallel SNLs F1 and F2 are generated to constrain the DOF of rotation of component 1 in the Y-direction and the DOF of rotation in the plane. The left end of components 2 and 3 are in contact with the rack point at B and C, respectively, moving in a vertical guide groove, resulting in collinear SNLs F3, F4 and F5, F6. It can be seen from Theorem 4 that the collinear CNLs each produce an over-restraint at B and C. The left ends of rods 2 and 3 are respectively in contact with the rack point at B and C through a spherical roller, moving in a vertical guide groove, resulting in collinear static normals F3, F4 and F5, F6. It can be seen from Theorem 4 that the collinear CNLs each produce an over-restraint at B and C.Figure 13
Lifting shear frame mechanism.One DOF of component 1 drives the movement of components to produce two branches, which act on component 8 together to produce two intersecting MNLsF7 and F8. Theorem 6 reveals that the intersecting MNLs restrict the freedom of movement of the constrained component. Based on the initial movement of component 1 with downward freedom of movement, the intersecting effective MNL constrains the freedom of movement of component 8 in the X-direction. Besides, two parallel SNLs F9 and F10 are generated since the frame and component 8 are connected by a kinematic pair. The movement DOF and plane of component 8 in the Y-direction are generated. The DOF of rotation within is restricted. The analysis of Theorem 9 normal line geometric theorem demonstrates that the effective intersecting dynamic normal line and the parallel SNL both restrict the freedom of movement of component 8 in the X-direction, and component 8 has an over-constraint.According to the geometric conditions, all components of a mechanism and the kinematic pair are symmetric along the axis of component 1 in the plane. As a result, the velocity components transmitted from component 6 and component 7 to component 8 in theX-direction are the same, implying velocity matching. In the process of movement, the geometric relationship of the normal lines on the parallel component does not change. Thus, the over-constraint has continuity and constitutes the virtual constraint: V = 3.The mechanism has 8 movable components,n = 8, 9 revolute pairs, 2 sliding pairs, and 4 higher pairs. The number of virtual constraints is V = 3. The DOFs can be obtained by substituting the formula:(7)F=3n−∑i=1mpi+V=3×8−2×2+9−1×4+3=1.The analysis of the above typical planar mechanisms with virtual constraints reveals that the position, nature, and quantity of the virtual constraints in the mechanism can be accurately analyzed through the geometric theorem of CNLs, and the DOF of a mechanism can be accurately calculated by substituting the calculation formula into it.


## 5.1. Analysis of the DOF of the Equal Diameter CAM Mechanism

Figure10 exhibits a simplified diagram of an equal-diameter CAM mechanism. Component 2 is the CAM, and the axis of the sliding pairs of component 3 is parallel at A and B. Component 2 generates collinear constraint normals F1 and F2 on component 3 at two points, O1 and O2, respectively. There are rotational DOFs in the plane based on the connection between component 2 and the frame through the revolute pairs at point O, implying that F1 and F2 are MNLs. According to Theorem 7, the two MNLs acting on component 3 are collinear. Hence, there is an over-constraint.Figure 10
The equal diameter cam mechanism.Based on the geometry relationship, the frame on both ends ofA and B by the prismatic pair connected to component 3 can be simplified into four parallel SNLs of F3, F4, F5, and F6 owing to the kinematic pairs of axial parallels. According to Theorem 5, any two parallel SNLs constrain the freedom of movement and rotation of component 3 in the X-direction, and the other two parallel SNLs are over-constraints.In the process of a mechanism’s movement, the positions and directions of parallel SNLsF3 and F4, F5, and F6 on the parallel component do not change, while the positions of dynamic normals F1 and F2 change but are always collinear. Hence, the over-constraints on the parallel component 3 are continuous. In this way, the three over-constraints on component 3 are virtual constraints, namely, V = 3.The mechanism has 2 movable components,n = 2, 2 higher pairs (each higher pair is limited to one DOF), 1 revolute pair (each revolute pair is limited to 2 DOFs), and 2 sliding pairs (each of which is limited to 2 DOFs, and the number of virtual constraints is V = 3). The DOFs can be obtained by substituting the following formula:(4)F=3n−∑i=1mpi+V=3×2−1×2−2×1+2+3=1.


## 5.2. Analysis of the DOF of the Press Mechanism

Figure11 presents a simplified diagram of the planar mechanisms of the press. The geometric conditions are that all components and the kinematic pairs of a mechanism are symmetrical along the line where component 1 is located in the plane. Component 8 is jointly restrained by the two rods 5 and 6 and the frame. Therefore, component 8 is a parallel component of a mechanism. Component 1 and the frame are connected by a sliding pair. According to the simplification of the plane kinematic pair normal line, component 1 has the function of parallel SNLs F1 and F2 in the X-direction. Theorem 2 suggests that the freedom of movement of component 1 in the X-direction and the rotation freedom in the plane are restricted, and only the freedom of movement is perpendicular to the normal line along the Y-direction.Figure 11
The press mechanism.One DOF of component 1 drives the movement of components 2 and 3 to produce two branches, which act on component 8 together to produce two intersecting MNLsF3 and F4. Theorem 6 reveals that the intersecting MNLs restrict the freedom of movement of the constrained component. Based on the initial movement of component 1 with downward freedom of movement, the intersecting effective MNL constrains the freedom of movement of component 8 in the X-direction. Besides, two parallel SNLs F5 and F6 are generated since the frame and component 8 are connected by a kinematic pair. The movement DOF and plane of component 8 in the X-direction are generated. The DOF of rotation within is restricted. The analysis of Theorem 9 normal line geometric theorem demonstrates that the effective intersecting dynamic normal line and the parallel SNL both restrict the freedom of movement of component 8 in the X-direction, and component 8 has an over-constraint.According to the geometric conditions, all components of a mechanism and the kinematic pair are symmetric along the axis of component 1 in the plane. As a result, the velocity components transmitted from component 5 and component 6 to component 8 in theY-direction are the same, implying velocity matching. In the process of movement, the geometric relationship of the normal lines on the parallel component does not change. Thus, the over-constraint has continuity and constitutes the virtual constraint: V = 1.The mechanism has 8 movable components,n = 8, 8 revolute pairs, and 4 sliding pairs. The number of virtual constraints is V = 1. The DOFs can be obtained by substituting the formula:(5)F=3n−∑i=1mpi+V=3×8−2×8+4+1=1.


## 5.3. DOF Analysis of the Planar 2-RRP Mechanism

The planar 2-RRP mechanism is exhibited in Figure12. The geometric conditions are CD∥EG and CE∥DG. According to the schematic diagram of a mechanism, component 2 has two branches. Thus, component 2 is the parallel component in the mechanism. Component 1 affects component 2 through rotation at point B. The judgment of the normal line type suggests that F3 is the freedom of movement of component 2 constrained by the SNL in the normal line direction, and F4 is an invalid dynamic normal line.Figure 12
The 2-RRP mechanism.The other branch acts on component 2 through components 4 and 3, and component 3 has the function of the SNLF7 to restrict the freedom of movement along rod 4. Additionally, there are two DOFs in the plane. On this basis, component 3 generates four effective horizontal action normal lines F5 and F6, F9 and F10 at points C and G in component 2 through the sliding pair. According to Theorem 6, the MNL has two over-constraints. Furthermore, the DOF of component 2 restricted based on component 3 is the same as the DOF of component 3. Therefore, component 2 is not constrained by the normal line of the flat behavior. The comprehensive analysis of Theorem 9 indicates that component 2 has two over-constraints.In the process of a mechanism’s movement, the effective MNLsF5 and F6, F9, and F10 are always parallel, and the constraints of the normal line of parallel component 2 do not change. Hence, the over-constraint on component 2 is the virtual constraint, and the number of virtual constraints of the whole mechanism is V = 2.The mechanism has 4 moving components,n = 4, 4 revolute pairs, and 2 kinematic pairs. The number of virtual constraints is V = 2. The DOFs can be obtained by substituting the following formula:(6)F=3n−∑i=1mpi+V=3×4−2×4+2+2=2.


## 5.4. DOF Analysis of the Lifting Shear Frame Mechanism

Figure13 shows a lifting shear frame that is hinged by multiple parallelograms and can obtain a larger telescopic stroke. It is a commonly used expandable telescopic mechanism for maintenance and warehouses. Component 1 and the rack are connected by a moving pair. It can be seen from Theorem 2 that two parallel SNLs F1 and F2 are generated to constrain the DOF of rotation of component 1 in the Y-direction and the DOF of rotation in the plane. The left end of components 2 and 3 are in contact with the rack point at B and C, respectively, moving in a vertical guide groove, resulting in collinear SNLs F3, F4 and F5, F6. It can be seen from Theorem 4 that the collinear CNLs each produce an over-restraint at B and C. The left ends of rods 2 and 3 are respectively in contact with the rack point at B and C through a spherical roller, moving in a vertical guide groove, resulting in collinear static normals F3, F4 and F5, F6. It can be seen from Theorem 4 that the collinear CNLs each produce an over-restraint at B and C.Figure 13
Lifting shear frame mechanism.One DOF of component 1 drives the movement of components to produce two branches, which act on component 8 together to produce two intersecting MNLsF7 and F8. Theorem 6 reveals that the intersecting MNLs restrict the freedom of movement of the constrained component. Based on the initial movement of component 1 with downward freedom of movement, the intersecting effective MNL constrains the freedom of movement of component 8 in the X-direction. Besides, two parallel SNLs F9 and F10 are generated since the frame and component 8 are connected by a kinematic pair. The movement DOF and plane of component 8 in the Y-direction are generated. The DOF of rotation within is restricted. The analysis of Theorem 9 normal line geometric theorem demonstrates that the effective intersecting dynamic normal line and the parallel SNL both restrict the freedom of movement of component 8 in the X-direction, and component 8 has an over-constraint.According to the geometric conditions, all components of a mechanism and the kinematic pair are symmetric along the axis of component 1 in the plane. As a result, the velocity components transmitted from component 6 and component 7 to component 8 in theX-direction are the same, implying velocity matching. In the process of movement, the geometric relationship of the normal lines on the parallel component does not change. Thus, the over-constraint has continuity and constitutes the virtual constraint: V = 3.The mechanism has 8 movable components,n = 8, 9 revolute pairs, 2 sliding pairs, and 4 higher pairs. The number of virtual constraints is V = 3. The DOFs can be obtained by substituting the formula:(7)F=3n−∑i=1mpi+V=3×8−2×2+9−1×4+3=1.The analysis of the above typical planar mechanisms with virtual constraints reveals that the position, nature, and quantity of the virtual constraints in the mechanism can be accurately analyzed through the geometric theorem of CNLs, and the DOF of a mechanism can be accurately calculated by substituting the calculation formula into it.


## 6. Conclusions

(1)
The DOF and its nature, over-constraint, and quantity can be accurately determined following the nature, quantity, and geometric relationship of the normal line acting on a component. Through the geometric relationship of the normal lines, the constraint nature of the constraint generated by the kinematic pair on the component can be accurately judged, and the degree of freedom of the mechanism can be analyzed intuitively from the constraint generation mechanism.(2)
The existence of over-constraint in a component is a necessary condition for the existence of virtual constraints. The CNLs acting on a component can continuously and invariably constrain a certain DOF of the component when the mechanism is moving. Meanwhile, over-constraint constitutes a virtual constraint. The comprehensive analysis of virtual constraints is transformed into the analysis of normal lines and geometric relationships, and the theoretical basis is sufficient, which avoids relying on empirical judgment.(3)
Based on the CNL, the geometric theorem is employed to judge the virtual constraints and their number in the mechanism. The method is rigorous, simple, and intuitive and can emerge from the difficulty of the analysis and calculation of the DOF of a mechanism. Thus, it lays a foundation for the application and promotion of teaching and engineering. It is easy for students and engineers to understand and master, laying a foundation for teaching and engineering application promotion.(4)
Further study the application of the method in the degree of freedom analysis of spatial parallel mechanism and coupling mechanism. A new method based on constrained normal lines and geometric relations is established for the analysis of spatial agency degrees of freedom.



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*Source: 3790060-2023-02-10.xml*