Comparative Study of TriVariant and Delta Three-Degree-of-Freedom Parallel Mechanisms for Aerial Manipulation

Abstract

The operational performance of robotic arms for multi-rotor flying robots (MFRs) has attracted growing attention in recent years. To explore new possibilities for aerial manipulation, this study investigates a novel parallel mechanism, the TriVariant, comprising one UP limb and two identical UPS limbs (2-UPS&UP). To evaluate its potential, we analyze its dimensional and kinematic characteristics and benchmark them against the widely adopted Delta robot, which is commonly integrated with unmanned aerial vehicles (UAVs). A prototype of the TriVariant is fabricated for experimental validation. Both analytical and experimental results reveal that, within a cylindrical task workspace characterized by a large diameter and moderate height, the TriVariant offers a more compact structure than the Delta robot, despite its slightly reduced dexterity. These findings highlight that the TriVariant is especially suitable for aerial manipulation in space-constrained environments where all limbs must be mounted beneath the UAV.

1. Introduction

Multi-rotor flying robots (MFRs), integrating unmanned aerial vehicles (UAVs) with robotic arms, possess exceptional flexibility and maneuverability. They are capable of performing tasks such as wall cleaning, power maintenance, fruit harvesting, non-destructive inspection, and crack repair in complex environments, including high altitudes and confined spaces [1,2,3,4]. By reducing labor costs and minimizing operational risks, MFRs exhibit significant potential for applications across agriculture, and industry.

In recent years, the operational performance of robotic arms for MFRs has garnered increasing research interest. Fixed or single-degree-of-freedom (1-DoF) robotic arms have been widely explored due to their structural simplicity and ease of modeling. For instance, a rigid manipulator arm [5,6] was integrated into a tilt-rotor UAV for inspection task. The 1-DoF arm presented in [7] achieves foldability and variable stiffness through an origami-inspired perpendicular folding mechanism. In [8], beyond an active rotational DoF provided by a servo motor, the manipulator incorporates a directional locking mechanism, allowing unidirectional movement of the end-effector while restricting the reverse motion, thereby mitigating impact forces during dynamic environmental interactions. However, the positioning accuracy of end effectors in fixed and 1-DoF robotic arms remains highly dependent on UAVs’ control precision, which imposes limitations on their practical applications.

Serial robotic arms are widely used in MFRs due to their structural flexibility, large workspace, and simple modeling [9]. For example, a quadrotor UAV has been equipped with a four-DoF robotic arm [10], while a hexarotor UAV integrates a six-DoF dual-arm system [11,12,13], both enabling large-amplitude movements. However, serial robotic arms still encounter challenges such as large mass, high inertia, insufficient rigidity, and low positioning accuracy. Consequently, their dynamic behavior may adversely affect the overall stability of MFRs.

Soft grippers have received considerable attention [14,15]. For instance, inspired by tendril plants and hook climbers, soft grippers driven by voltage-induced phase transitions of low-boiling-point liquids have been developed to achieve coiling deformations, eliminating the need for complex grasping strategies or high-precision positioning [16]. Similarly, drawing inspiration from octopus arms and elephant trunks, researchers have designed logarithmic spiral-shaped robots that replicate this geometric form across scales. These robots employ cable-driven actuation to enable controlled curling and uncurling motion [17]. MFRs equipped with soft grippers are capable of adaptively grasping objects of various shapes and sizes in unstructured environments. However, improving the positioning accuracy of soft grippers remains a key challenge.

Parallel manipulators, known for their high precision, stiffness, and speed, have been employed in MFRs to enhance positioning accuracy [18,19]. Most existing studies [20,21,22,23] utilize Delta robots [24,25,26,27] mounted beneath UAVs to perform tasks such as data acquisition, sample collection, and 3D printing. Other works [28,29,30,31] place Delta robots laterally on UAVs to enable interactions with vertical surfaces. In these configurations, UAVs are responsible for coarse positioning across large spatial areas, enabling fast global navigation and broad adjustments, while Delta robots execute fine positioning for high-precision operations within localized regions. However, the considerable number of links in the Delta robot may hinder its dexterous performance and pose challenges for accurate dynamic modeling. In addition to Delta robots, a six-DoF parallel manipulator with a 6-RUU configuration, comprising six legs, each with a driven revolute joint and two universal joints, has been developed and mounted directly on a UAV. However, its structural rigidity and load-bearing capacity remain insufficient for carrying grippers and performing physically interactive tasks [32]. Moreover, a 6-DoF hybrid cable-driven parallel manipulator featuring a central elastic rod has been integrated with a UAV for contact-based inspection of floating offshore wind turbines. Despite its compact design, this system currently lacks the ability to exert adequate force on the environment during operation [33].

Although the Delta robot remains mainstream in aerial manipulation, it raises a key question: is there a parallel mechanism that is both compatible with the Delta robot and suitable for aerial applications? In this paper, we turn our attention to the TriVariant [34,35,36,37] composed of one UP limb, two identical UPS limbs (2-UPS&UP), and a 2-DoF head on the moving platform. For simplicity, we use the term “TriVariant” to denote the 2-UPS&UP parallel mechanism. Similar to the Delta robot, the TriVariant features high dynamic performance, structural rigidity, and a favorable cylindrical workspace-to-footprint ratio. It has been successfully applied in high-speed machining tasks such as milling, drilling, deburring, welding, and assembly within the aerospace and automotive industries. To evaluate its suitability for aerial manipulation, this study conducts a comparative analysis of the TriVariant and Delta robot in terms of dimensional characteristics and kinematic performance. Additionally, a TriVariant prototype is fabricated for trajectory tracking evaluation and integrated with a UAV platform to validate its capabilities in aerial manipulation tasks.

The remainder of this paper is organized as follows. Section 2 details the forward and inverse kinematics of the TriVariant. Section 3 presents the forward and inverse kinematics of the Delta robot. Section 4 provides a comparative analysis of the two mechanisms in terms of dimensional characteristics and kinematic performance. Section 5 describes the trajectory tracking experiments of the TriVariant and its integration with a UAV platform for aerial manipulation. Section 6 provides a detailed discussion of this work, and Section 7 concludes the paper.

2. Kinematics of the TriVariant

In this paper, we aim to develop a MFR by integrating the TriVariant with a UAV, as shown in Figure 1a. The kinematic structure of the TriVariant is illustrated in Figure 1b. The moving platform and the fixed base are denoted by equilateral triangles $∆$A₁A₂A₃ and $∆$B₁B₂B₃, with side lengths of a and b, respectively. The i-th limb (i = 1, and 2) is a UPS limb, connecting the fixed base via a U joint and the moving platform via an S joint. The third limb is a UP limb, connecting the fixed base with a U joint at one end and perpendicularly fixing to the moving platform at the other. All rotational axes of the three U joints lie within the $∆$B₁B₂B₃ plane. A local Cartesian coordinate frame, B_j-x_jy_jz_j (j = 1, 2, and 3), is set up at the base where the origin of the frame is coincident with the point B_j; the x_j-axis lies within the $∆$B₁B₂B₃ plane and is perpendicular to the line segment B₁B₂; the z_j-axis is normal to the $∆$B₁B₂B₃ plane; and the y_j-axis follows the right-hand rule. The designed task workspace of the TriVariant, denoted by W_t, is cylindrical with a relatively large radius R₁ and moderate height h₁. Further, to achieve axisymmetric kinematic performance relative to the x₃-z₃ plane within W_t, a global Cartesian coordinate frame, O-XYZ, is defined. the X-axis is collinear with the x₃-axis; the Y-axis lies within the $∆$B₁B₂B₃ plane and is parallel to the y₃-axis; the Z-axis aligns with the central axis of W_t; and the origin O is offset from the point B₃ with a distance of e. The vertical distance from the base to the upper boundary of W_t is denoted by H₁.

2.1. Forward Kinematics

The positions of points A₁, A₂ and A₃ are denoted by

$${\mathrm{A}}_1={\left[\begin{array}{ccc}{x}_1& y_1& z_1\end{array}\right]}^{\mathrm{T}}$$

(1)

$${\mathrm{A}}_2={\left[\begin{array}{ccc}{x}_2& y_2& z_2\end{array}\right]}^{\mathrm{T}}$$

(2)

$${\mathrm{A}}_3={\left[\begin{array}{ccc}{x}_3& y_3& z_3\end{array}\right]}^{\mathrm{T}}$$

(3)

According to the structure of the TriVariant, eight geometric constraints are obtained:

$$\left|{\mathbf{A}}_{\mathbf{1}}{\mathbf{A}}_{\mathbf{2}}\right|=\left|{\mathbf{A}}_{\mathbf{1}}{\mathbf{A}}_{\mathbf{3}}\right|=\left|{\mathbf{A}}_{\mathbf{2}}{\mathbf{A}}_{\mathbf{3}}\right|=a$$

(4)

$$\left|{\mathbf{A}}_i{\mathbf{B}}_i\right|=q_i i=1,2,3$$

(5)

where q_i is the length of the axial axis of the i-th limb (i = 1,2, and 3).

$${\mathbf{A}}_{\mathbf{3}}{\mathbf{B}}_{\mathbf{3}}\perp {\mathbf{A}}_{\mathbf{1}}{\mathbf{A}}_{\mathbf{3}}$$

(6)

$${\mathbf{A}}_{\mathbf{3}}{\mathbf{B}}_{\mathbf{3}}\perp {\mathbf{A}}_{\mathbf{2}}{\mathbf{A}}_{\mathbf{3}}$$

(7)

It can be observed that the number of unknown parameters for points A₁, A₂ and A₃exceeds the number of geometric constraints by one. As a result, given the lengths of the three limbs q_i, the solutions for A₁, A₂ and A₃ is not unique. For more details, please refer to [37].

2.2. Inverse Kinematics

The vector of B₃A₃ is given by

$${\mathit{r}}_3=q_3{\mathit{w}}_3$$

(8)

$${\mathit{r}}_3={\mathit{b}}_i+q_i{\mathit{w}}_i-{\mathit{a}}_i, i=1,2$$

(9)

where w_i is the unit vector of the axial axis of the i-th limb; a₁, a₂, b₁, and b₂ are the vectors of A₃A₁, A₃A₂, B₃B₁, B₃B₂, respectively.

The norms of Equations (8) and (9) are given by

$$q_3=\left|{\mathit{r}}_3\right|$$

(10)

$$q_i=\left|{\mathit{r}}_3+{\mathit{a}}_i-{\mathit{b}}_i\right|, i=1,2$$

(11)

2.3. Jacobian Matrix

Rewriting Equations (8) and (9), it yields

$${\mathit{w}}_3={\displaystyle \frac{{\mathit{r}}_3}{q_3}}$$

(12)

$${\mathit{w}}_i={\displaystyle \frac{{\mathit{r}}_3+{\mathit{a}}_i-{\mathit{b}}_i}{q_i}},i=1,2$$

(13)

The velocity mapping function of the TriVariant is given by [35]

$${\dot{\mathit{r}}}_3=\mathit{J}\dot{\mathit{q}}$$

(14)

where $\dot{ \mathit{q}}$ is the velocity of three prismatic joints of three limbs; $\dot{\mathit{r}}$ is the velocity of A₃; J is the Jacobian matrix, and

$$\dot{\mathit{q}}={[\begin{array}{c}{\dot{q}}_1\end{array} \begin{array}{c}{\dot{q}}_2\end{array} {\dot{q}}_3]}^{\mathrm{T}}$$

$$\mathit{J}={({[{\mathit{w}}_1 {\mathit{w}}_2 {\mathit{w}}_3]}^T-\frac{1}{q_3}[{{\mathit{w}}_1^T{\mathit{w}}_3{\mathit{a}}_1 {\mathit{w}}_2^T{\mathit{w}}_3{\mathit{a}}_2 0]}^T)}^{-1}$$

3. Kinematics of the Delta Robot

The Delta robot is composed of a fixed base, a moving platform, and multiple identical limbs, each consisting of an upper arm and a forearm. As shown in Figure 2a, the upper arm is connected to the fixed base and the forearm via revolute joints. The forearm forms a parallelogram structure, with its four links connected by spherical joints. One of the forearm links is either rigidly attached to the moving platform or connected to it through a revolute joint.

Figure 2b illustrates the schematic diagram of the Delta robot. The fixed base and the moving platform are denoted by circles A₁A₂A₃ and C₁C₂C₃ centered at O and P, respectively, with radii r₁ and r₂. The upper arms are denoted by A_iB_i with a length of L₁, and the forearms are denoted by B_iC_i with a length of L₂. To simplify the model, each limb is translated by a distance of r₂ along the direction of C_iP. Such that, A_i is shifted to D_i; B_i to E_i, and C_i coincides with P. Consequently, the radius of the new circle D₁D₂D₃ is given by r = r₁ − r₂.

3.1. Forward Kinematics

A global Cartesian coordinate frame O-XYZ is set up with its origin at O; the X-axis aligns with OA₁; the Z-axis is normal to the A₁A₂A₃ plane; and the Y-axis is determined by the right-hand rule. The vector of point P is denoted by

$$\mathit{p}={\left[\begin{array}{ccc}{X}_P& Y_P& Z_P\end{array}\right]}^T$$

(15)

According to the structure of the Delta robot, three geometric constraints are obtained:

$$\begin{array}{cc}\left|{\mathit{c}}_{\mathit{i}}\right|=L_2,& i=1,2,3\end{array}$$

(16)

where c_i is the vector of E_iP. As the number of unknown parameters of point P is equal to the number of geometric constraints, the position of point P can be calculated, and the workspace of the Delta robot is obtained.

3.2. Inverse Kinematics

The angle between OD_i and the X-axis is denoted by α_i. The angle between OD_i and D_iE_i, is denoted by θ_1i. The vector of the point E_i is expressed as

$$\begin{array}{c}{\mathit{e}}_i=(X_{Ei},Y_{Ei},Z_{Ei}{)}^{\mathrm{T}}={\mathit{a}}_{\mathit{i}}+{\mathit{b}}_{\mathit{i}}={}^{\mathbf{0}}\mathit{R}_{\mathit{i}}\left(\left[\begin{array}{c}r\\ 0\\ 0\end{array}\right]+L_1\left[\begin{array}{c}\cos {\theta }_{1i}\\ 0\\ \sin {\theta }_{1i}\end{array}\right]\right), i=1,2,3\end{array}$$

(17)

where a_i and b_i are the vectors of OD_i and D_iE_i, respectively, and ⁰R_i is a rotation matrix:

$${}^{\mathit{0}}\mathit{R}_{\mathit{i}}=\left[\begin{array}{ccc}\cos {\alpha }_i& -\sin {\alpha }_i& 0\\ \sin {\alpha }_i& \cos {\alpha }_i& 0\\ 0& 0& 1\end{array}\right]$$

The length of E_iP is calculated by

$${\left(X_{Ei}-X_P\right)}^2+{\left(Y_{Ei}-Y_P\right)}^2+{\left(Z_{Ei}-Z_P\right)}^2=L_2^2$$

(18)

Rewriting Equation (18), it yields

$$M_{1i}\cos {\theta }_{1i}+M_{2i}\sin {\theta }_{1i}=M_{3i}$$

(19)

where

$$\begin{array}{l}{M}_{1i}=2L_{1}r-2L_1\cos {\alpha }_i{X}_P-2L_1\sin {\alpha }_i{Y}_P\\ M_{2i}=-2L_1{Z}_P\\ M_{3i}=L_2^2-L_1^2-{\left(X_P-r\cdot \cos {\alpha }_i\right)}^2-{\left(Y_P-r\cdot \sin {\alpha }_i\right)}^2-Z_P^2\end{array}$$

Equation (19) is solvable if and only if

$$M_{1i}^2+M_{2i}^2\ge M_{3i}^2$$

(20)

3.3. Jacobian Matrix

According to the structure of the Delta robot shown in Figure 2b, we have

$$\mathit{p}={\mathit{a}}_i+{\mathit{b}}_i+{\mathit{c}}_i$$

(21)

Differentiating Equation (21), it yields

$$\dot{\mathit{p}}={\dot{\mathit{a}}}_i+{\dot{\mathit{b}}}_i+{\dot{\mathit{c}}}_i={\mathit{\omega }}_{bi}\times {\mathit{b}}_i+{\mathit{\omega }}_{ci}\times {\mathit{c}}_i$$

(22)

where ω_bi and ω_ci are the angular velocities of b_i and c_i, respectively. Multiplying both sides of Equation (22) by a unit vector $\widehat{\mathit{c}}$_i, it yields

$${\mathit{c}}_i\cdot \dot{\mathit{p}}={\mathit{c}}_i\cdot ({\mathit{\omega }}_{bi}\times {\mathit{b}}_i)=({\mathit{b}}_i\times {\mathit{c}}_i)\cdot {\mathit{\omega }}_{bi}=({\mathit{b}}_i\times {\mathit{c}}_i)\cdot {\widehat{\mathit{\omega }}}_{bi}\cdot {\theta }_{1i}$$

(23)

where $\widehat{\mathit{\omega }}$_bi is the unit vector of ω_bi.

$${\widehat{\omega }}_{bi}={}^{\mathit{0}}\mathit{R}_{\mathit{i}}\cdot {\left[\begin{array}{ccc}0& 1& 0\end{array}\right]}^T={\left[\begin{array}{c}-\sin {\alpha }_i \cos {\alpha }_i 0\end{array}\right]}^T$$

(24)

Equation (23) can be rewritten as

$${\mathit{J}}_{\mathit{p}}\dot{\mathit{p}}={\mathit{J}}_{\mathit{\theta }}\dot{\mathit{\theta }}$$

(25)

where

$${\mathit{J}}_{\mathit{p}}={\left[\begin{array}{ccc}{\mathit{c}}_1& {\mathit{c}}_2& {\mathit{c}}_3\end{array}\right]}^{\mathrm{T}}$$

$${\mathit{J}}_{\mathit{\theta }}=\left[\begin{array}{ccc}({\mathit{b}}_1\times {\mathit{c}}_1)\cdot {\widehat{\mathit{\omega }}}_{b1}& 0& 0\\ 0& ({\mathit{b}}_2\times {\mathit{c}}_2)\cdot {\widehat{\mathit{\omega }}}_{b2}& 0\\ 0& 0& ({\mathit{b}}_3\times {\mathit{c}}_3)\cdot {\widehat{\mathit{\omega }}}_{b3}\end{array}\right]$$

Equation (25) can be further rewritten as follows

$$\dot{\mathit{p}}=\mathit{J}\dot{\mathit{\theta }}$$

(26)

where J is the Jacobian matrix, and

$$\mathit{J}={\mathit{J}}_{\mathit{p}}^{-1}{\mathit{J}}_{\mathit{\theta }}$$

4. Performance Comparison of the TriVariant and the Delta Robot

4.1. Dimension

To compare the performance of the TriVariant and Delta robot, their design parameters must first be defined. Three key considerations are taken into account during this process. First, both robots are required to operate within an identical cylindrical task workspace, characterized by a relatively large diameter and moderate height. Second, the diameter of their moving platforms must be identical to ensure a fair comparison. Finally, the design parameters of both robots are optimized to ensure they operate in their best configurations for achieving the designed workspace prior to performance evaluation.

It is worth noting that different optimization approaches, such as the minimum dimension criterion [26] and motion/force transmission indices [27], can significantly affect the selection of design parameters. To ensure generality and fairness, the design parameters of the Delta robot are determined by scaling down one of the most widely adopted industrial models, the ABB IRB 360-1/800. In contrast, the design parameters of the TriVariant robot are based on the optimized results presented in [35], with their relationships defined in Equation (27). A summary of the design parameters used for comparison is provided in

Table 1. Design parameters of the TriVariant and Delta robot for comparison (mm).

TriVariant	R1	h1	H1	e	a	b	q3min	q3max	q1min/q2min	q1max/q2max
	200	100	279	107.5	39	137.5	279	488	262	458
Delta	R2	h2	H2	L1	L2	r1	r2
	200	100	300	117.5	400	100	22.5

.

$$\left\{\begin{array}{l}{R}_1=R_2, h_1=h_2\hfill \\ a=\sqrt{3}{r}_2, b=R_1/1.455, e=0.782b\hfill \\ q_{3\min }=H_1, q_{3\max }=\sqrt{(R_1+e)+(H_1+h_1)}\hfill \\ {\displaystyle \frac{q_{i\max }-q_{i\min }}{q_{i\min }}}\le 0.75,i=1,2,3\hfill \end{array}\right.$$

(27)

Based on the design parameters listed in Table 1, the reachable workspace of the TriVariant is illustrated as grey dots in Figure 3a, encompassing the designed cylindrical workspace with a radius of 200 mm and a height of 100 mm, highlighted in blue. Similarly, the reachable workspace of the Delta robot is shown in Figure 3b with the same designed workspace indicated. These results confirm that both robots achieve identical task workspaces under the given design parameters. Notably, the TriVariant features a smaller base radius and a shorter vertical distance from its base to the workspace compared to the Delta robot. Moreover, all three limbs of the TriVariant are located beneath its base, whereas the upper arms of the Delta robot may extend above its base. These structural characteristics indicate that the TriVariant exhibits a more compact structure compared to the Delta robot.

4.2. Dexterity

The condition number $k$ of the Jacobian matrix is one of the most suitable local performance measures for evaluating the dexterity between the joint variables and the reference point of the moving platform [38]. To compare the kinematic performance of the TriVariant with the Delta robot, the condition number of the Jacobian of both robots is evaluated throughout the designed workspace W_t using the design parameters given in Table 1.

$$\kappa \left(\mathit{J}\right)=\parallel \mathit{J}\parallel \cdot \parallel {\mathit{J}}^{-1}\parallel$$

(28)

Since $k$ varies with changes in configuration of the system, two global conditioning indices, the mean value $\overline{k}$ and standard deviation $\tilde{k}$ of $k$ in the designed workspace W_t, are also used as performance indices.

$$\overline{\kappa }={\displaystyle \frac{{\int }_{W_t}\kappa d{W}_t}{W_t}}$$

(29)

$$\tilde{\kappa }={\displaystyle \frac{\sqrt{{\int }_{W_t}{(\kappa -\overline{\kappa })}^{2}d{W}_t}}{W_t}}$$

(30)

The local and global conditioning indices $k$, $\overline{k}$, and $\tilde{k}$ of the TriVariant and the Delta robot are summarized in

Table 2. The condition numbers and singular values between the TriVariant and the Delta robot.

	$\overline{\mathit{k}}$	$\tilde{\mathit{k}}$	${\mathit{k}}_{\mathbf{m}\mathbf{i}\mathbf{n}}$	${\mathit{k}}_{\mathbf{m}\mathbf{a}\mathbf{x}}$	σ
TriVariant	6.935	0.954	5.144	10.179	[0.577, 5.886]
Delta robot	2.952	0.296	2.533	4.213	[42.503, 247.027]

. The spatial distribution of the condition number $k$ of the TriVariant within the designed workspace W_t is shown in Figure 4a, and that of the Delta robot is shown in 4b. Similarly, the condition numbers of the TriVariant and the Delta robot across the top, middle, and bottom layers of W_t are presented in Figure 4c,d, respectively. As illustrated, the TriVariant exhibits higher mean and standard deviation values of the condition number compared to the Delta robot, indicating slightly reduced dexterity within the designed workspace.

In addition to the condition number, the maximum and minimum singular values of the Jacobian matrix at a given position i, denoted by σ_max,i and σ_min,i, are employed as indices to identify regions of the designed workspace where the end-effector can achieve either high velocities or fine pose resolution [39]. The singular values of a Jacobian matrix J are defined as the non-negative square roots of the eigenvalues of the symmetric matrixJ^TJ.

$${\sigma }_{k,i}=\sqrt{{\lambda }_{k,i}({\mathit{J}}^{\mathrm{T}}\mathit{J})} k=1,2,3$$

(31)

where λ_k,i denotes the k-th eigenvalue. The maximum and minimum singular values are obtained by

$${\sigma }_{\max ,i}=\max \left({\sigma }_{1,i},{\sigma }_{2,i},{\sigma }_{3,i}\right)$$

(32)

$${\sigma }_{\min ,i}=\min \left({\sigma }_{1,i},{\sigma }_{2,i},{\sigma }_{3,i}\right)$$

(33)

In fact, the singular values of the Jacobian quantify how joint velocities are mapped into end-effector velocities. A larger singular value corresponds to a direction in which end-effector velocity is amplified, while a smaller singular value corresponds to a direction where motion is reduced or lost. in the limiting case, a singular value of zero indicates a singular configuration [40].

For both the TriVariant and the Delta robot, the singular values σ_max,i and σ_min,i are evaluated throughout the designed workspace W_t using the design parameters given in Table 1. The distributions of the maximum and minimum singular values of the TriVariant are shown in Figure 5a,b, respectively, while those of the Delta robot are illustrated in Figure 5c,d. As summarized in Table 2, the singular values σ of the TriVariant range from 0.577 to 5.886., i.e., all σ_max,i values are less than or equal to 5.886, and all σ_min,i values are greater than or equal to 0.577. By contrast, the singular values σ of the Delta robot lie within a much larger interval, from 42.503 to 247.027. These results indicate that the Delta robot achieves higher end-effector velocities, while the TriVariant offers finer end-effector resolution. Neither robot exhibits a singular configuration within the designed workspace.

5. Trajectory Tracking Experiments of the TriVariant

To evaluate the trajectory tracking performance of the TriVariant, a 479 g prototype was constructed according to the design parameters listed in Table 1. The prototype employed three linear DC motors (Hoodland, IP60; Hoodland Technology Co., Ltd., Wenzhou, China) integrated with Hall-effect displacement transducers. Each motor provides a maximum linear speed of 30 mm/s, an output force of 20 N, and a self-locking force of 20 N. Each displacement transducer offers a resolution of 38 pulses per millimeter of motor displacement. To control motor positions, the pulses generated by displacement transducers during motion were counted, and the motors were stopped once the target positions were reached, with an allowable error of up to three pulses. A control system consisting of a Jetson Nano (NVIDIA, Santa Clara, CA, USA), an Arduino Uno (Arduino, Milan, Italy), and two L298N drivers (STMicroelectronics, Geneva, Switzerland) was implemented, as illustrated in Figure 6, and the prototype was mounted on a wooden test platform. No rated load was applied to the TriVariant; instead, a marker was attached to the end of the moving platform, and its motion was recorded using the FZMotion motion tracking system. All experiments were conducted indoors to eliminate airflow interference.

A circular trajectory with a radius of 0.8R₁ was defined for the end-effector of the TriVariant. The input commands for the three linear DC motors were computed through inverse kinematics. The motor trajectories and corresponding tracking errors during experiments are shown in Figure 7. Results demonstrate that the motors effectively follow the input signals, with tracking errors remaining below 0.5 mm. The end-effector tracking errors of the TriVariant are presented in Figure 8 and

Table 3. MAE of the TriVariant during trajectory tracking.

Direction	X	Y	Z	3D
MAE (mm)	3.437	3.736	3.909	7.719

. The experimental results indicate that the TriVariant can follow the desired trajectory with a mean absolute error (MAE) of 7.719 mm. The observed errors are primarily attributed to two factors. First, the Hall-effect displacement transducers are not perfectly integrated with the linear DC motors, causing some pulses to be unrecorded during motor movement. As a result, the motors may not stop precisely at their target positions, leading to reduced accuracy. Second, the universal and spherical joints used in assembling the TriVariant prototype exhibit clearance, which reduces the stability of the moving platform. Specifically, when an external force is applied, the platform may deviate by up to 4 mm even if all three linear DC motors are locked. In addition, the instability of the wooden test platform shown in Figure 6b contributes slightly to the overall tracking errors.

A MFR was developed by integrating the TriVariant with a UAV. The UAV’s roll, pitch, yaw, and thrust were controlled via a Pixhawk, while the TriVariant was managed by a Jetson Nano. Figure 9 illustrates snapshots of the TriVariant-integrated MFR operating at various positions, highlighting its suitability for aerial manipulation, particularly in scenarios where all limbs must be mounted beneath the UAV.

6. Discussions

This paper focuses on comparing the TriVariant and the Delta robot in terms of dimension and dexterity, while other aspects such as stiffness [41] and motion/force transmissibility [42] could be further investigated for a more comprehensive evaluation. The TriVariant prototype was assembled using low-cost linear DC motors, Hall-effect displacement transducers, and joints, with each motor (including a built-in transducer) costing approximately USD 20. To enhance accuracy, future prototypes could employ high-resolution linear motors and low-clearance joints. Moreover, the payload capacity of the current prototype was not evaluated in experiments; this could be analyzed through force transmissibility in conjunction with the maximum output force of the linear motors.

Both the TriVariant and the Delta robot have three DoFs. Conventionally, three actuators are employed, each mounted on one limb, to adjust the position of the moving platform, which is also the configuration adopted in our performance analysis. To further improve force transmissibility, dynamic performance, and energy efficiency, future work could explore kinematic redundancy by adding actuated joints or kinematic chains [43,44,45]. In addition, the high-speed motion of the TriVariant may induce instability in the MFR. Therefore, optimizing its design parameters and developing coordinated trajectory planning strategies for both the UAV and the TriVariant merit further investigation [46].

7. Conclusions

This paper presented the kinematic modeling and performance comparison of the TriVariant and the Delta robot. Analytical results showed that, within a cylindrical task workspace characterized by a relatively large diameter and moderate height, the TriVariant offers a more compact structure than the Delta robot, despite its slightly reduced dexterity. A prototype of the TriVariant was developed, and experimental results reveal that it can track the desired trajectory with a MAE of 7.719 mm. Furthermore, A MFR was constructed by integrating the TriVariant with a UAV, demonstrating TriVariant’s suitability for aerial manipulation tasks in which all limbs are required to be positioned beneath the UAV.

Future work will focus on optimizing the design parameters of the TriVariant and integrating it with a two-DoF head for directional adjustment. In addition, advanced control algorithms will be developed to enhance its environmental interaction capabilities, including object picking, pulling, and pushing.