19.2 - Kinematic Analysis

While the compliant finger mechanism can be structurally modeled as a five-bar system, the torsional spring at one joint alters the expected motion behavior. Before the finger contacts an object, the spring is stationary, allowing the finger to be analyzed as a standard four-bar linkage. After contact, the spring flexes and the motion of one link is constrained by the object, creating a new four-bar configuration with different grounded links. The kinematic analysis is therefore divided into these two four-bar configurations.

Mechanism Stages	Torsional Spring	Finger Motion

Mechanism Stages	Torsional Spring	Finger Motion
Stage 1	Stationary, unflexed	Full finger collectively bends
Stage 2	Acts as a joint, flexes	Only distal segment bends

Stage 1: Four-Bar Mechanism Analysis

Initially, the compliant finger acts as a four-bar mechanism as shown in the schematic drawn below. In this configuration, the torsional spring remains stationary, and the full linkage system moves collectively, making the entire finger bend. This first four-bar analysis applies up to the moment the finger, specifically link 3 in this schematic, contacts an object.

Figure 1. Kinematic Diagram of 1st Four-Bar, Not to Scale

Mobility Calculation

The Gruebler equation for this first four-bar mechanism is calculated as seen below:

Gruebler Equation: M = 3(L - 1) - 2J₁- J₂

L = 4 links, J₁ = 4 full joints, J₂ = 0 half joints

M = 3(4 - 1) - 2(4) - 0 = 9 - 8 = 1 Degree of Freedom

This results in 1 DOF which is expected from the standard four-bar setup.

Additionally, because we do not anticipate any links needing to rotate a full 360 degrees, the Grashof condition does not necessarily need to be met. The following calculation demonstrates that the four-bar mechanism does meet the Grashof condition and is thus a Class I Kinematic Chain.

Grashof Condition: S + L < P + Q

Figure 2. Link Length and Geometry of 1st Four-Bar Kinetic Diagram

Link Lengths, as defined in Figure 2, are used to analyze the S + L < P + Q Grashof condition:

S = shortest link = 13.2 mm

L = longest link = 55 mm

P = remaining link = 22.4 mm

Q = remaining link = 50 mm

Therefore, 68.2 < 72.4, confirming that the four-bar is a Grashof mechanism. Since the shortest link is the coupler, the mechanism is classified as a double-rocker. However, as mentioned previously, full rotation is not required for this application, as the compliant finger is only intended to achieve a limited range of motion for grasping. Accordingly, the physical design is constrained to prevent complete 360° rotation and instead operate within a functional and more useful grasping range.

Position Analysis

Utilizing the geometry defined in Figure 2, a position analysis is conducted to generate the position profile of point B, as shown in Figure 3. The local x–y coordinate frame is oriented at an offset of 116.6° relative to the global X–Y frame (Figure 2), and the mechanism is evaluated over an input range of motion from 0 to −30° with respect to the global frame. This four-bar system is in the crossed configuration.

This position analysis is essential for understanding the resulting trajectory of the finger, allowing us to verify that the motion follows a suitable path for grasping. We are interested in point B for this analysis because when the finger comes into contact with an object, point B becomes a new grounded joint for the second four-bar configuration. Thus, understanding the kinematics of point B will help us understand how the finger transitions between the two four-bar setups. In particular, it helps ensure that the compliant finger closes in a controlled and predictable manner around an object, rather than following an undesirable or inefficient path.

Figure 3. 1st Four-Bar Position Profile at Point B

Velocity Analysis

Similarly, a velocity analysis is performed to generate the velocity profile shown in Figure 4, assuming a constant input angular velocity of 1 rad/s or roughly 9.55 revolutions per minute.

This analysis provides insight into how quickly point B moves throughout the motion cycle and highlights variations in speed across the range of motion. Understanding the velocity profile is important for evaluating how smoothly the finger engages with an object, as excessive speeds could lead to instability or reduced control during grasping. On the other hand, a slower, more uniform motion is generally more desirable for precision.

Figure 4. 1st Four-Bar Velocity Profile at Point B

Acceleration Analysis

An acceleration analysis is also conducted to produce the acceleration profile presented in Figure 5. This analysis captures how the velocity of point B changes over time, providing further insight into the grasping stability. Evaluating the acceleration profile therefore helps ensure that the mechanism operates smoothly and safely within its intended range of motion.

Figure 5. 1st Four-Bar Acceleration Profile at Point B

Mechanical Advantage

The mechanical advantage of the mechanism, defined as the ratio of output force F4 to input force F2, is analyzed to evaluate how effectively the four-bar transmits force, as shown in Figure 6. This is useful for the compliant finger application, as it indicates how input actuation forces are applied on the finger. Understanding the mechanical advantage across the range of motion helps ensure that sufficient force can be applied for grasping while also avoiding overly large output forces that could damage delicate objects.

Figure 6. 1st Four-Bar Mechanical Advantage

Animation

Lastly, the animation below depicts the four-bar animation in motion with respect to the global X-Y coordinate frame. The input angle changes from 0 to -30 degrees to reflect an useful grasping range for the first four-bar configuration. Additionally, the velocity vectors are shown to help visualize the motion profile.

Figure 7. Animation of 1st Four-Bar

Stage 2: Four-Bar Mechanism Analysis

Once the finger contacts an object, some joints become constrained, effectively grounding different links. As the input continues to actuate, the torsional spring flexes, causing motion in other links. This sequential bending occurs exclusively at the “distal joint” of the finger, allowing it to wrap around smaller objects. The schematic below shows this second four-bar configuration, and the following analysis covers this phase until full closure.

Figure 8. Kinematic Diagram of 2nd Four-Bar, Not to Scale

Mobility Calculation

Similarly to the first four-bar mechanism, the Gruebler equation yields the same result for this second four-bar configuration. This is shown below in the following calculation, resulting in 1 DOF:

Gruebler Equation: M = 3(L - 1) - 2J₁- J₂

L = 4 links, J₁ = 4 full joints, J₂ = 0 half joints

M = 3(4 - 1) - 2(4) - 0 = 9 - 8 = 1 Degree of Freedom

The Grashof condition can be checked in the following calculations, and as expected, this four-bar system does not met the condition.

Grashof Condition: S + L < P + Q

Figure 9. Link Length and Geometry of 2nd Four-Bar Kinetic Diagram

Link Lengths, as defined in Figure 9, are used to analyze the S + L < P + Q Grashof condition:

S = shortest link = 10 mm

L = longest link = 50 mm

P = remaining link = 15 mm

Q = remaining link = 41.7 mm

Therefore, 60 < 56.7 is not true, and the four-bar is a non-Grashof mechanism.

In the context of our project, this four-bar mechanism being a non-Grashof system is acceptable as we do not need any links to rotate 360 degrees for the adaptive grasping behavior.

Position Analysis

Analysis for this 2nd four-bar can vary depending on the size of the object the mechanism is trying to grasp. The size of the object changes the mechanism’s point of contact with the object, which affects the location of the ground point between L1 and L4.

For analysis purposes, we assume that the finger-mechanism makes contact with an object when it becomes parallel with the global X-axis. Geometrically, this also causes the local x-y axis to be equivalent to the global X-Y axis. Additionally, this four-bar system is in the crossed configuration.

The allowable range of motion for the four-bar is roughly -13 to -29 degrees in the global X-Y coordinate frame. These values are calculated in Figure 10. The minimum range of motion happens when the mechanism first makes contact with the object, whilst the torsional spring’s angle is still at its equilibrium angle of 120 degrees. On the other hand, the maximum range of motion occurs when the torsional spring has been fully compressed to 0 degrees, causing links 3 and 4 to align. Thus, this represents a toggle point configuration we want to avoid.

Figure 10. Range of Motion for 2nd Four-Bar

With the geometry and assumptions defined in Figure 9, the position profile of Point P, located at the end of the distal finger segment, is shown in Figure 11. In this second four-bar configuration, Point P represents the fingertip and is therefore the most relevant point for evaluating grasping performance. Tracking its position provides direct insight into how the finger wraps around smaller objects after the proximal link becomes effectively grounded.

Figure 11. 2nd Four-Bar Position Profile at Point P

Velocity Analysis

Similarly, a velocity analysis was performed to evaluate how motion occurs through the mechanism, with results shown in Figure 12. A constant input angular velocity of 1 rad/s was assumed for the input link. The resulting velocity profile of Point P characterizes the speed of the fingertip during the wrapping motion, which is critical for ensuring smooth and controlled contact with objects.

Figure 12. 2nd Four-Bar Velocity Profile at Point P

Acceleration Analysis

An acceleration analysis was additionally conducted, with the resulting profile shown in Figure 13. This analysis highlights regions where the fingertip experiences rapid changes in motion, which can influence contact forces and overall grasp stability.

Figure 13. 2nd Four-Bar Acceleration Profile at Point P

Mechanical Advantage

The mechanical advantage of the system, defined as the ratio of output force F4 at the distal link to the input force F2, is shown in Figure 14. This reflects how effectively input forces are transmitted to the fingertip in the second configuration. A higher mechanical advantage in this stage is desirable, as it enables the distal segment to apply sufficient force at Point P to securely grasp smaller objects.

Figure 14. Mechanical Advantage for 2nd Four-Bar

Animation

Finally, the animation illustrates the kinematic behavior of the mechanism in this second four-bar configuration by showing both link positions and corresponding velocity vectors throughout the motion. The distal finger segment is visualized with Point P at the fingertip, allowing for a clear representation of how the finger moves and interacts with objects during the grasping process.

Figure 15. Animation of 2nd Four-Bar

Torsional Spring Analysis

The torsional spring is a critical component of the compliant finger mechanism, as it enables the transition between the two four-bar configurations. During the first stage of motion, the spring should remain effectively stationary, allowing the mechanism to behave as a rigid four-bar system. In the second stage, the spring must be sufficiently compliant so that the motor can easily overcome its resistance, allowing the distal segment to continue rotating and wrap around the object. This balance between stiffness and flexibility is essential for achieving the desired adaptive grasping behavior.

To estimate an appropriate spring stiffness, the torsional spring constant k can be approximated using relationships based on the spring’s geometry and material properties. Specifically, k depends on the wire diameter, mean coil diameter, number of active coils, and the elastic modulus of the material (in this case, stainless steel 304). Using the equation and assumed values below, based on the springs we ordered, we can determine our approximate spring constant.

Governing Equation for Spring Constant: k = (E * d^4) / (10.8 * D * N)

where:

E = Young’s modulus of material
d = wire diameter
D = mean coil diameter
N = number of coils

Our values based on the springs from our Draft BOM (see 19.3 - Initial Prototype):

For 304 stainless steel, we assume E is roughly 193 GPa or 193 * 10^9 Pa
Wire diameter: d = 1 mm = 0.001 m
Mean coil diameter: D = 5 mm = 0.005 m
Number of coils: N = 6

Plugging in values:

k = (193*10^9)(0.001)^4 / (10.8 * 0.005 * 6) = Approximately 0.6 Nm / rad

While the physical springs have not yet been integrated into the prototype, we believe this estimated spring constant value falls within a reasonable range based on comparable systems. For example, torsional springs used in everyday devices, such as clothespins, typically range from approximately 0.01 to 0.1 Nm/rad, while compliant elements in robotic linkages often fall between 0.2 and 0.8 Nm/rad. In contrast, heavy-duty or automotive torsional springs have stiffness values several orders of magnitude larger, which would be unsuitable for this application.

Using the estimated spring constant, we can estimate the amount of torque required to bend the joint and move the distal finger for different angular displacements. For example, to bend a torsional spring with a resting angle of 120 degrees to 60 degrees, we can approximate the torque as follows:

T = k * delta_theta

k = 0.6 Nm/rad
delta_theta = new position - resting position = 60 - 120 degrees = -1.047 radians

Torque = -0.63 Nm (Negative sign indicates the torque is resisting motion)

This provides insight into our actuator selection and overall system design moving forward. Because an assortment kit of torsional springs has been ordered, future work will include systematic testing across springs with varying wire diameters and rest angles. This will allow the best spring to be selected to achieve our desired performance of the compliant finger mechanism. More is discussed in “19.3 - Initial Prototype.”