Understanding the degree of freedom in robots

We have already learned about the different components that make up an industrial robot or a robotic arm. We have also discussed different types of robots used in industry. One of the most important terms used in the context of robots and their design is degree of freedom. In this article, we will talk about the degree of freedom in robots. Why it is so important and how it is calculated.

What is the degree of freedom?
The degree of freedom is an extremely important factor in robotics that is used to define the physical motion capabilities of a robot. A robot is essentially a combination of multiple mechanisms where each mechanism is formed by a set of links and joints. As already discussed, there are different types of joints used in building a robot. The most common lower pair joints include the revolute joint, prismatic joint, helical joint, cylindrical joint, spherical joint, and universal joint. Each joint has its own degree of freedom. Revolute, helical, and prismatic joints have 1 DoF, cylindrical and universal joints have 2 DoF and spherical joints have 3 DoF. When the degree of freedom is determined for a joint, it generally refers to the number of axes the joint offers motion to attached links.

When multiple joints are connected to form a robotic arm or industrial robot, the resultant degree of freedom is different. In the context of a robot, the degree of freedom is the number of independent variables or parameters that define the motion or configuration of the robot. DoF represents the number of ways a robot can move or position itself in its working space. The more the degree of freedom, the more flexible and adaptable the robot. A robot with high DoF can make more complex movements and perform a variety of tasks. Higher DoF, however, comes at the expense of more cost, complexity, and control challenges.

How to calculate the degree of freedom of a robot?
Determining the degree of freedom of a robot may not be that easy. For certain types of robots, it is quite straightforward. For example, a Cartesian robot is always 3-DoF. A Scalar robot is 4-DoF. Though for other types of robots and robot designs, it is not that simple. If each joint of the robot is independent and has no constraints and dependencies, the total DoF of the robot is the sum of the DoF of all joints. For example, if a robot has two revolute joints and two prismatic joints, all independent without any dependencies, the total DoF of the robot is 4 (2 + 2), with each joint adding one degree of freedom. Generally, joints are not all independent and have some constraints applied to them. The dependencies of joints further complex the equation.

Figure 1.

Each type of joint has a certain number of independent constraints. For example, the number of independent constraints between two planar rigid bodies for both revolute and prismatic joints is 2. The number of independent constraints between two spatial rigid bodies for revolute and prismatic joints is 5. The number of independent constraints between two spatial rigid bodies for helical, cylindrical, universal, and spherical joints is 5, 4, 4, and 3 respectively. The degree of freedom of a specific joint is the difference of the number of independent constraints from the sum of freedoms of bodies. By that calculation, revolute, helical, and prismatic joints have 1 DoF, cylindrical and universal joints have 2 DoF and spherical joints have 3 DoF.

The degree of freedom of independent joints is summarized in Table 1.

Table 1.

As each robot has a different mechanism, the calculation for the degree of freedom could be different. A common way of determining the degree of freedom of a robot is the Kutzbach criterion and Chebychev-Grubler criterion, well known as Grubler’s Formula. The Kutzbach criterion applies only to the degree of freedom of mobility of a plane mechanism. According to the Kutzbach criterion, for a planar or spatial mechanism to be mobile i.e. having a degree of freedom greater than 0, the number of constraint equations (C) must be greater than or equal to the difference in the number of links (L) from the number of independent kinematic parameters (P).

C >= P-L for any planar or spatial mechanism to be mobile.

According to the Chebychev-Grubler criterion, the degree of freedom for the mechanism is given by the following equation (Equation 1).

Equation 1.

Where:

m = degrees of freedom of a single body/link, which equals 3 for a rigid body moving in a plane and equals 6 for a rigid body moving in three-dimensional space.

N = Number of links/bodies including ground

J = Number of joints

Ci = Number of constraints provided by joint i

fi = Number of freedoms provided by joint i

On applying the Chebychev-Grubler criterion to a planar mechanism, the equation for DoF is:

DoF = 3(L-1) – 2J – H

Where:

L = Number of links

J = Number of binary joints

H = Number of higher pairs

Let us apply the Chebychev-Grubler criterion to some planar mechanisms having no higher pairs for calculating the degree of freedom.

For a closed chain three-bar mechanism as shown in Figure 2.

Figure 2.

DoF = 3(3-1)-2×3 = 0

For a closed chain four-bar mechanism as shown in Figure 3.

Figure 3.

DoF = 3(4-1)-2(4) = 1

For a closed chain five-bar mechanism as shown in Figure 4.

Figure 4.

DoF = 3(5-1)-2(5) = 2

For a closed chain five-bar mechanism with two ternary joints as shown in Figure 5.

Figure 5.

DoF = 3(5-1)-2(6) = 0

For a closed chain six-bar mechanism with four ternary joints as shown in Figure 6.

Figure 6.

DoF = 3(6-1) – 2(8) = -1

The Grubler’s Formula can be applied to only those mechanisms that have 1 DoF joints i.e., the joints forming the robotic system are revolute, prismatic, or helical. It does not apply to mechanisms having 2-DoF or 3-DoF joints. The formula can also be applied to open-chain mechanisms. For example, an open-chain mechanism having 3 revolute joints and one prismatic joint is shown in Figure 7.

Figure 7. Open-chain mechanism having 3 revolute joints and one prismatic joint.

There are four links/bodies connected by four joints each joint having 1-DoF, then by Grubler’s Formula

DoF = 3 (4-1-4) + (1+1+1+1) = 1

The Grubler’s Formula is not applicable when joints are not independent. For example, consider the following mechanism.

On applying Grubler’s equation to the above mechanism (Figure 8), we get a DoF of 0.

Figure 8. Open-chain mechanism having 3 revolute joints and one prismatic joint with Grubler’s equation applied.

DoF = 3 (3-1-3) + (1+1+1) = 0

But the above mechanism can be seen to clearly have a degree of freedom equal to 1. Because the middle joint in the above mechanism is not independent, Grubler’s Formula does not apply to it.

Finding the degree of freedom for simple robot designs is easy, as there are more non-independent joints and more dependencies, the determination of the degree of freedom of a mobile robot gets more and more complex.

For simplicity, we can reduce most of the robotic mechanical designs into open-chain and closed-chain mechanisms as shown in Figure 9.

Figure 9.

Provided the robot has only 1-DoF joints only like revolute, prismatic, and helical joints, the degree of freedom of the robot can be easily calculated using Grubler’s Formula. As the robot involves non-independent joints, dependencies, and 2-DoF & 3-DoF joints, the calculations get more complicated.

Why DoF is important?
Each robot has a different mechanical structure. With the same links, a robot can be arranged using different joints and in many ways. The degree of freedom of a robot impacts many areas and plays a key role in defining the robot’s performance. First, DoF determines the range and complexity of motions that a robot can achieve. It is a key factor in deciding how robots will navigate and manipulate objects in different configurations and working environments. A higher DoF robot can access and manipulate objects from many different angles and orientations enabling it to perform more complex tasks.

Secondly, DoF is an important consideration in determining the task adaptability of the robot. Different tasks require different ranges of motion, approach angles, and orientations. Simpler tasks can be accomplished by lower DoF tasks, while more complex tasks require higher DoF. Plus, the degree of freedom of a robot is again an important factor in determining the working space of the robot. A higher DoF robot not only has a larger working space, but it can also utilize the available working space more efficiently. Therefore, the degree of freedom is important for determining the reachable working space and accessibility to the objects in the reachable workspace.

The degree of freedom also determines the kinematic redundancy and redundancy resolution of the robot. The degree of freedom of a robot designed for an intended task must not be higher than required. Even unnecessarily high DoF can pose obstacles in the movement of a robot, make it less energy efficient, and hamper its stability. The degree of freedom must be enough to enhance the kinematic redundancy of the robot so that the robot can effectively repeat movements in an energy-optimized manner maintaining good stability and repeatability. The robot design must be optimized so its motion and posture are as per specific objectives.

The degree of freedom of a robot also affects the control and planning of the robot. As the degree of freedom of a robot is increased, more complex control algorithms and techniques are required to plan, coordinate and control multiple degrees of freedom effectively. The planning and control of a robot having multiple degrees of freedom involve addressing challenges like path planning, trajectory generation, collision avoidance, and joint coordination.

Hence, the degree of freedom of a robot plays an important role in defining its range and reach, adaptability, repeatability, stability, performance, complexity, and efficiency.