Kinematic Analysis of a Serial-Parallel Hybrid Mechanism and its Application to a Wheel-Legged Robot

A three-DOF (degree-of-freedom) serial-parallel hybrid mechanism is proposed that consists of a 2-UPS+U component (UPS: universal-/prismatic-/spherical-joint chain and U: universal joint) and an R+RPS component (R: revolute joint and RPS: revolute-/prismatic-/spherical-joint chain). This mechanism can provide high precision and a large workspace, which indicates its applicability to wheel-legged robots. To verify our proposal, a mobility analysis showed that the mechanism simultaneously achieved three rotational movements. Then, the inverse/forward position analysis, velocity analysis, acceleration analysis, and stiffness analysis were carried out. Finally, a prototype of the mechanism was fabricated and its position error was found to be <1.8% despite the large workspace. These results demonstrate the effectiveness of our proposal and provide a new approach for designing new mechanisms for wheel-legged robots.


I. INTRODUCTION
Wheel-legged mobile robots (WLMRs) have attracted much attention because of their adaptability to various environments [1]. A WLMR is a hybrid system of a legged robot and a wheeled robot. A WLMR moves quickly and efficiently on wheels in flat road conditions, but can also function in complex road conditions, such as rugged and muddy roads, and pass through obstacles with legs.
WLMRs can be roughly divided into three types according to design concept. (1) A wheeled structure with a legged suspension, e.g., ANYmal and ATHLETE [2]- [13]. The wheel and the leg are generally connected in series, with the wheel set at the end of the leg. (2) A legged/wheeled alternative structure, e.g., Halluc IIx [14] and NOROS [15]. To improve the speed on flat roads, driving/passive wheels are mounted The associate editor coordinating the review of this manuscript and approving it for publication was Tao Wang . on the leg. Here, depending on road conditions, the two modes can be freely switched with a legged/wheeled conversion device. (3) A simple combination of wheeled and legged robots, e.g., Handle [16]- [18].
The majority of the above WLMRs adopt serial mechanisms, which offer advantages such as easily arranged joints, large workspaces, and simple controllability, but their low stiffness makes them less accurate in both motion and capacity. In contrast, parallel mechanisms are capable of carrying heavy loads with high motion accuracy but suffer from small workspaces, strong motion restriction caused by the other components, and control difficulties. Clearly, these advantages and disadvantages are complementary; this indicates that combining them in a hybrid mechanism design would provide the advantages of both serial and parallel mechanisms. Thus, the serial-parallel hybrid mechanism (SPM) has emerged as a choice for the innovative design of robot mechanisms. VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ To date, some trials have been conducted to apply SPMs to robots. Thornton [19] proposed an assembly robot, the GEC Tetrabot, which serially combines a 3-DOF wrist manipulator and a 3-DOF parallel mechanism. This design created a relatively large workspace compared to conventional parallel mechanisms. Giuseppe and Marco [20] devised a serial/parallel robot for surgical use by serially connecting a SCARA robot and a 3-DOF parallel manipulator. Cha et al. [21] developed a robot for road maintenance based on SPMs, which connects a 6-UPS (UPS: universal-/prismatic-/spherical-joint chain) parallel manipulator to a 3-DOF serial frame. Gao et al. [22] designed a quadruped walking robot based on SPMs. The leg of this robot serially includes a 3-RRR (R: rotation joint) parallel mechanism and an R joint, used as the hip and knee joints, respectively. Hou et al. [23] proposed a 5-DOF manipulator based on an RP-(2-RRU/1-RUU) mechanism. Peidró et al. [24] developed a humanoid robot with a serial/parallel hybrid structure; its leg incorporates two parallel mechanisms that are connected in series. Zhang and Gao [25] combined two parallel mechanisms in series to form a robot, obtaining a large workspace and high accuracy. Lu et al. [26] developed a serial-parallel hybrid manipulator with an upper 3RPS (RPS: revolute-/prismatic-/spherical-joint chain) parallel mechanism and a lower 3SPR mechanism and used it to build a hand with three fingers. Wang et al. [27] devised an SPM as the leg of a quadruped robot, with a four-bar parallel mechanism and five-bar mechanism in series.
The above results demonstrate that SPMs can be categorized into two types: (1) Two parallel mechanisms that are serially connected and (2) serial and parallel mechanisms that are serially connected [28]. The first type has been rarely reported, as it is difficult to construct theoretical models for analysis and control. In contrast, the second type has been broadly studied, as the parallel mechanism can achieve high position accuracy, while the serial mechanism can improve the dexterity and enlarge the workspace.
We propose an SPM [29] and apply it as a leg for a wheellegged robot. This SPM essentially consists of a 2-UPS+U and an R, and it belongs to type (2). In [29] the workspace of the SPM was found, showing that the mechanism has a larger workspace compared with the workspace of conventional parallel mechanism. Thus, as the PM in this study is structurally similar to the previous one, it should inherently exhibit the merit of large workspace. However, the stiffness of the entire mechanism is insufficient because of the serial portion. To mitigate this problem, in this study, we improve the previous mechanism by adding an RPS chain and forming a new mechanism, namely, a (2-UPS+U)&(R+RPS). Here, using an RPS chain is capable of forming a double R-pairs connection type, which enhances the stiffness of the entire mechanism and consequently improves the motion accuracy. In addition, because the new mechanism is structurally similar to the previous mechanism, it can provide a large workspace. These features allow us to infer that our proposal would provide a good candidate mechanism for wheel-legged robots.
In this paper, the configuration of the (2-UPS+U)& (R+RPS) is first introduced, and its mobility is demonstrated. Subsequently, the kinematics are theoretically analyzed, and the results are verified through simulation. Finally, a mechanism prototype is fabricated, and its performance is experimentally evaluated to assess the feasibility of our proposal.

II. CONFIGURATION AND DOF ANALYSIS A. CONFIGURATION
Figs. 1(a) and (b) illustrate the mechanism's schematic and 3D model, respectively. An RPS kinematic chain is added between the limbs l 1 and l 2 . The P pair is an active joint, the R and S pairs are passive joints, and the rotation of l 2 is achieved with the actuator l 03 .
To facilitate the following analysis, we have named the symbol of each part of the mechanism, as shown in Table 1.

B. DOF ANALYSIS
First, the DOF of the (2-UPS+U)&(R+RPS) mechanism is analyzed to clarify its mobility. Because the configuration is a spatial SPM, the DOF is obtained with the modified Kutzbach-Grübler formula [30], [31]: (1) In the mechanism, the order d equals 6, the number of components n equals 9, the number of kinematic pairs g equals 11, and the sum of the DOFs of all the kinematic pairs f i equals 20. The UPS chain's DOF is 6, and the kinematic chain does not have any constraint on the mechanism, so the 2-UPS+U does not have a redundant constraint. In the R+RPS, the analysis shows that this part has 1 DOF and 5 effective constraints, and the total number of constraints is 6 (the R pair has 5 constraints, and the RPS chain has 1 constraint), so the redundant constraint v is 1 (6-5 = 1); see the analysis below for details. In addition, there is no local DOF in the mechanism (indicating that η = 0). According to (1), the mechanism's DOF is 3.
Then, the mobility of the mechanism is analyzed based on screw theory [32], [33]. In the 2-UPS+U, the U pair between the frame and the upper limb l 1 can be equivalently treated as two revolute joints, so there are two screws: ¡ S 1 and ¡ S 2 . On the other hand, in the constraint wrench of this part, obtained based on the reciprocal screw theory, there are four constraint wrenches: ¡ S r 1 , ¡ S r 2 , ¡ S r 3 and ¡ S r 4 . Thus, the kinematic screws and constraint wrenches of the 2-UPS+U can be expressed as Here, the four constraint wrenches constrain three movements in the x-, y-and z-axes and the rotation in the z-axis, respectively. Thus, the 2-UPS+U retains 2 DOFs, i.e., the rotations in the x-and y-axes.
In the R+RPS, the R pair has one screw ¡ S 1 and five constraint wrenches ¡ S r 1 , ¡ S r 2 , ¡ S r 3 , ¡ S r 4 and ¡ S r 5 ; the kinematic screws and constraint wrenches of the R pair can be expressed as Meanwhile, the RPS chain has five screws, ¡ S " 1 , ¡ S " 2 , ¡ S " 3 , ¡ S " 4 and ¡ S " 5 , and one constraint wrench ¡ S "r , as shown in Fig. 2. Thus, the kinematic screws and constraint wrenches of the RPS chain can be expressed as where [0 e h] T is the position coordinate of the S pair in the C 3 − x 3 y 3 z 3 coordinate system. By (3) and (4), we can see that rank ( ¡ S r and ¡ S r 5 , so ¡ S "r is a redundant constraint. Thus, the R+RPS retains 1 DOF, i.e., the rotation in the x k -axis. In summation, the 3 DOFs of the (2-UPS+U)&(R+RPS) mechanism are the rotations in the x-, y-and x k -axes.

III. KINEMATIC ANALYSIS A. POSITION ANALYSIS 1) FORWARD POSITION ANALYSIS
A forward position analysis of the (2-UPS+U)&(R+RPS) mechanism is implemented in the coordinate systems shown in Fig. 1(a). O-xyz is a global coordinate system, α and β are the rotation angles in the x-and y-axes, and γ is the rotation angle in the x k -axis in a moving coordinate system K -x k y k z k . To clarify the position of the end point (P) in O-xyz, it can be calculated with the following steps: Express the position vector of point K in O-xyz as where O K R = Rot(Y , β)Rot(X , α) and l 1 = [00 l 1 ] T is the vector of limb l 1 .
Determine the relationship between O-xyz and K − x k y k z k and express it with a transformation matrix Through the coordinate transform, the point p position in K − x k y k z k is obtained as VOLUME 8, 2020 where K P R = Rot(X K , γ ) and l 2 = [0 0 l 2 ] T is the vector of limb l 2 .
Determine the relationship between the moving coordinate system P-x p y p z p and K-x k y k z k by using a homogeneous transformation matrix: Multiply (6) by (8) and build the relationship between Oxyz and P-x p y p z p : Calculate the position vector of the end point P in From (10), the relationships between the point P position (p x , p y , p z ) and the three rotation angles α, β and γ are obtained. Next, the relationships between α, β and γ and the lengths of the electric push rods l 01 , l 02 , and l 03 need to be analyzed to determine the actuating method.
The central positions of the three U pairs in O-xyz are The central position of each point of the upper limb in The central position of each point of the lower limb in K-x k y k z k are Accordingly, the lengths of l 01 and l 02 can be determined from α and β with the following equation: where · represents the 2-norm of the vector.
Using (11)- (14), l 01 and l 02 can be expressed with geometrical parameters as (15), shown at the bottom of this page, where sα = sinα, cα = cosα, sβ = sinβ, and cβ = cosβ. (15) provides an approach to obtain the angle α and β by adjusting l 01 and l 02 . In particular, where In addition, where The relationship between the rotation angle γ and the length of the electric push rod l 03 can be obtained as (18), shown at the bottom of the next page.
Then, the expression of γ can be obtained from (17) Through the adjustment of l 01 , l 02 and l 03 , the designed values of α, β and γ and the consequently designed point P position are achievable with (16), (17), and (19). The above process is the forward position analysis. Next, the inverse position is analyzed to provide a basis for trajectory planning.

2) INVERSE POSITION ANALYSIS
Through inverse position analysis, a method to adjust the l 01 , l 02 and l 03 can be established to obtain the desirable position (p x , p y , p z ) of point P in O-xyz. First, the values of α, β and γ should be solved from the point P position. In Fig. 1(a By substituting α, β and γ obtained from (20) into (15) and (18), the lengths of three electric push rods l 01 , l 02 and l 03 can be determined.

B. VELOCITY ANALYSIS 1) JACOBIAN MATRIX
The Jacobian matrix is a commonly used concept for velocity analysis, and it reflects the mapping relationship between the operating space and joint space.
In the (2-UPS+U)&(R+RPS) mechanism, the velocity in the operating space is referred to as the linear velocity of point P, and the velocity in the joint space includes two rotational angular velocitiesα andβ of the upper limb's U-pair and the rotational angular velocityγ of joint K . Therefore, the Jacobian matrix of the mechanism is the mapping function between the angular velocitiesα,β andγ and the linear velocity of point P.
By acquiring the partial derivatives of p x , p y and p z alternately by α, β and γ , whose relationships are given in (10), we obtain After transforming (21), (22), and (23) into matrix form, we obtain where J va ∈ R 3×3 is the Jacobian matrix of the (2-UPS+U)&(R+RPS) mechanism. Further, since the changes in α, β and γ are caused by l 01 , l 02 , and l 03 , the actual inputs are the variation in the lengths of the three electric push rods. Therefore, the generalized Jacobian matrix between the input velocitiesl 01 ,l 02 andl 03 and the linear velocity of point P can be constructed.  Fig. 3(a) shows that, in the quadrilateral OA 1 B 1 C 1 , OC 1 ⊥B 1 C 1 , OA 1 , OC 1 and B 1 C 1 are fixed values, and A 1 B 1 represents the length of electric push rod, l 01 . From the geometric relationship, O OC 1 is the forward rotational angle α of the mechanism, so we obtain Similarly, the quadrilateral OA 2 B 2 C 2 shown in Fig. 3 By timely deriving β and l 02 , we obtaiṅ Similarly, by deriving γ and l 03 in the time domain, we obtaiṅ Through the above analysis, the transfer function J al ∈R 3×3 between the angular velocitiesα,β andγ and the linear velocitiesl 01 ,l 02 andl 03 of the three electric push rods are where . Therefore, the Jacobian matrix between the point P velocity and the linear velocities of three electric push rods can be obtained as where J vl ∈ R 3×3 .

2) CENTROID VELOCITY OF EACH COMPONENT
Once the Jacobian matrix is obtained, we can analyze the centroid velocity of each component of the mechanism. First, we analyze the centroid velocity of the limb l 1 . Since the limb l 1 , incorporates multiple parts and is unevenly distributed, the centroid does not locate at the geometrical center. Here, we define the centroid's initial position as l c1 in the O-xyz. The centroid velocity includes linear and angular velocities, where the angular velocity is the sum of angular velocitiesα andβ. Thus, Since the angular velocities of all the points on l 1 are the same, (33) is applicable to the other points on l 1 . Based on the influence coefficient method, the linear velocity of l 1 can be obtained as where O P c1 (= O K Rl c1 ) is the position vector of the centroid of l 1 and S() represents the calculation of the skew-symmetric matrix of a vector.
Second, we analyze the centroid velocity of the limb l 2 . Here, we define the centroid's initial position as l c2 in the K-x k y k z k . Through matrix transformation, the angular velocity of the centroid of l 2 is ω c2 = ω α + ω β + ω γ = r α ·α + r β ·β + r γ ·γ =   cos β0 cos β 010 where The linear velocity of l 2 can be obtained as ) is the position of the centroid of l 2 , K is the initial position of K-x k y k z k in O-xyz, Third, we analyze the centroid velocity of three kinetic limbs, i.e., two UPS chains and one RPS chain, because their large weights can affect the dynamic properties.
In the (2-UPS+U)&(R+RPS) mechanism, the S pairs of the two UPS chains are connected to the limb l 1 , so their central points B i (i = 1, 2) of the S pairs have the same velocities, equal to that of l 1 : The linear velocity ofB i (i = 1, 2) is expressed as where l B i are the positions of the central points B i (i = 1, 2) of the S pairs. Ref. [34] reported that the velocity of the UPS chain depends on that of its U pair. Therefore, we establish two local coordinate systems, i.e., A i − x i y i z i (i = 1, 2) at the central point A i of the U pairs shown in Fig. 4. Since the three U pairs are in the same direction, the A i -x i y i z i are parallel to the O-xyz. Meanwhile, the rotation direction of α i and β i are the same as that of the U pair of l 1 . Thus, where l 0i are the positions of the central points B i (i = 1, 2) of the S pairs in A i − x i y i z i ,α i andβ i are the angular velocities of the U pair, and r αi and r βi are the directional vectors of the two joint angles α i and β i , respectively. Combining (38), (39) and (40), we obtain where In A i − x i y i z i , the velocities of the centroids of the UPS chains can be obtained as where l c 0i are the positions of the UPS chains' centroids in Here, since we initially rotate l 1 by an angle α in the x-axis and then rotate by an angle β in the y-axis, some of the components in (44) can be defined as Finally, the centroid velocity of the RPS chain is analyzed. The velocity of this RPS chain relies on the velocity of the R pair. Since the R pair is connected with l 1 , the angular velocity of the center point of the R pair is the same as limb l 1 . The angular velocity of the R pair in O-xyz is In O-xyz, the angular velocity of the RPS chain's centroid can be obtained as VOLUME 8, 2020 where α 3 is a function of the length of the RPS chain and can be obtained as By deriving α 3 and l 03 , in the time domain, we obtaiṅ The linear velocity of the RPS chain's centroid is where O P c 03 is the position of the RPS chain's centroid in O-xyz, and l c 03 is the position of the UPS chain's centroid in C 3 − x 3 y 3 z 3 shown in Fig. 2.

C. ACCELERATION ANALYSIS
After obtaining the Jacobian matrix, we analyze the point P acceleration by partially differentiating (24) in the time domain and obtain the following equation: where H = [ H Px H Py H Pz ] T is the Hessian matrix, which is obtained by partially differentiating each component in Jacobian matrix J va by α, β and γ alternately: Similarly, the relationship between the angular acceleration of each joint and the acceleration of each electric push rod is that where H = [ H α H β H γ ] T is the Hessian matrix, obtained by partially differentiating each component in Jacobian matrix J al by l 01 , l 02 and l 03 alternately. The results are as follows:

D. INVERSE VELOCITY AND INVERSE ACCELERATION ANALYSIS 1) INVERSE VELOCITY ANALYSIS
When the velocity v x v y v z T of point P is known, the speed l 01l02l03 T of three electric push rods can be calculated as where J lv = J −1 al J −1 va is the inverse Jacobian matrix.

2) INVERSE ACCELERATION ANALYSIS
When the acceleration of the endpoint is known, there are two steps used to solve the acceleration of the three electric push rods. The first step determines the relationships between the velocities and accelerations of each joint and those of point P, and the second step determines the relationships between the joints and the electric push rods. First, by differentiating (20), we obtain where J av ∈ R 3×3 is the matrix showing the relationships between the joint velocities and the point P velocity.
Through a partial derivation of (56), we obtain  α β γ where H av = H α H β H γ T is the Hessian matrix obtained by partially differentiating each component in Jacobian matrix J av by p x , p y and p z alternatively: Then, by differentiating (15) and (18), we obtain where J la ∈ R 3×3 is the matrix indicating the relationships between the joint velocities and the electric push rod velocities.
By implementing a partial derivation of (59), we obtain  l 01 l 02 l 03 where H la = H l 01 H l 02 H l 03 T is the Hessian matrix, obtained by partially differentiating each component in Jacobian matrix J la by α, β and γ alternatively:

IV. STIFFNESS ANALYSIS
When loads are applied to the robot, its end effector produces undesirable displacement, which directly affects the robot's position accuracy. Since the robot's accuracy is mainly affected by the stiffness, it would be important to analyze the stiffness. In particular, the relationship between the driving system's stiffness and the end effector's stiffness should be clarified. It is worth mentioning that, to simplify the modeling, driven system is regarded as a flexible part, while the others as rigid ones.
Representing the force vector of the driving joint in the three limbs with f = [f 1 f 2 f 3 ] T , the corresponding deformation with q = [ q 1 q 2 q 3 ] T , and the stiffness coefficient with k = diag(k 1 , k 2 , k 3 ), respectively, we have Meanwhile, the force vector of the leg-end is F = [F 1 F 2 F 3 ] T , and corresponding deformation is Q = [ Q 1 Q 2 Q 3 ] T , and K is the stiffness Jacobian matrix, then their relationship is that By rewriting the Jacobian matrix into the differential form: According to the principle of virtual work, we obtain that, Substituting (62), (63) and (64) into (65), we obtain that There are many methods for evaluating the stiffness of a mechanism, e.g., the determinant, the condition number, as well as the eigenvalue distribution of the stiffness Jacobian matrix. To clearly show the influence of the stiffness of driving joints and the error of the Jacobian matrix on the stiffness Jacobian matrix of the leg-end, we use the condition number of the stiffness Jacobian matrix to evaluate the stiffness of this mechanism. The results obtained by numerical analysis are shown in Fig. 5. In the workspace, the condition numbers of stiffness Jacobian matrix at z = 500, 600, 700, and 800 mm are shown in Figs. 5(a), (b), (c), and (d), respectively.
As shown in Fig. 5, the condition number of the stiffness Jacobian matrix is satisfactorily small in the whole working space; this indicates that the robot is capable of obtaining high stiffness and avoiding large force error of the leg-end induced by the errors of the joints' force and Jacobian matrix.

V. SIMULATION ANALYSIS
To verify the effectiveness of the above analysis, a simulation is implemented. First, the mechanism is modeled with SolidWorks (ver. 2016). Then, the model, where constraints and actuating conditions are considered, is analyzed with ADMAS (ver. 2013). Finally, the simulated results are compared with the theoretical values.
Prior to simulation, the initial position of point P is set at [−400 0 −450] T and three electric push rods' velocities are expressed as The displacements of the three rods are given as Using MATLAB (ver. R2014a), the results obtained through theoretical analysis and simulation are processed. The displacement, velocity and acceleration of point P are shown in Figs. 6, 7, and 8, respectively. It can be observed that the theoretical values (indicated in black lines) agree with the simulated results (red lines); this shows that the theoretical analysis is effective.

VI. EXPERIMENT
A prototype of the (2-UPS+U)&(R+RPS) mechanism was fabricated. The photo of the prototype is shown in Fig. 9. The prototype consists of a frame, an upper leg, a lower leg, three electric push rods (SC0, Shidaichaoqun Corp., Beijing, China), a wheel, and a controller. The upper and lower legs are 450 and 400 mm, respectively.
To evaluate the accuracy, it is necessary to measure the position error. Currently, commonly used calibration methods include direct and indirect approaches. The direct method tracks the positions of the reference point of the measured object, while the indirect method calculates the parameter VOLUME 8, 2020 changes, as measured with length/angle sensors, through forward kinematics [35]. In comparison, the results obtained by the direct method include the assembly errors and control errors of each limb, and the overall accuracy is theoretically higher than the indirect method; thus, the direct method is selected. The instrument used in the experiment was a Vicon MX motion capture system (OML Corp., Oxford, UK). Four cameras tracked the markers in real time. The experimental setup is shown in Fig. 10. The experimental processes are listed as follows: (1) The prototype was fixed to the calibration stand, and reflective balls, used as markers, were bonded to  the prototype. The position of the prototype was adjusted to ensure that the position of each ball could be captured with three cameras.
(2) The origin of the spatial coordinate system was determined with a T-shaped calibration frame.
(3) A trajectory was planned and the electric push rods were actuated to enable the movement of the prototype to fit the trajectory. The cameras were used to record the real positions of the mechanism. The experiments were carried out 5 times.
(4) The positions obtained from the 5 experimental runs were averaged as the trajectory curve and the error of each point was obtained by comparing them with the planned curve.
The planned trajectory curve and the experimental trajectory of point P is shown in Fig. 11. The maximum deviation in the y-axis displacement between the actual and planned  curves was 0.39 mm, corresponding to a maximum error of 1.8%. The spatial curve was entirely smooth, particularly in the xoz plane; this implied that the robot could move smoothly when the robot walked forward. Because the movement of the prototype in the y direction was not sufficiently stable, the trajectories in the xoy and yoz planes were not smooth. In particular, the curves oscillated in the y range between 20 and 22 mm. This result was likely caused by jitter when the electric push rod reached the end.
In addition, the mechanism could move smoothly in the x-axis range of >400 mm and the z-axis range of >300 mm, both of which implied its suitability as the leg of a wheellegged robot.
Finally, the stiffness and position accuracy of the current mechanism are compared with those of the mechanism reported in [29]. Observably, the stiffness is approximately twice higher for the current than for the previously-reported mechanism. On the other hand, the z-axis accuracy of the current mechanism is 4% higher than that of the previous one. These results prove the advantages of the current mechanism, i.e., high stiffness and high accuracy, as shown in Table 2.

VII. CONCLUSION
In this paper, a 3-DOF serial-parallel mechanism is proposed, and its feasibility and potential are proved through theoretical analyses and experiments. First, by analyzing the mobility of the mechanism, the configuration has 3 DOFs, which include rotations around the x-, y-and x k -axes. Then, kinematics analyses and simulations are conducted, and the results well agree with the theoretical values; this indicates that the theoretical derivations of the position, Jacobian matrix, velocity, acceleration, and inverse kinematics of the mechanism are effective. Finally, the mechanism prototype is fabricated, and its performance is evaluated. The experiments show that the mechanism offers high motion accuracy and a large workspace, both of which imply its suitability as a leg for wheel-legged robots.
In this study, the kinematic properties of our mechanism are systematically studied, and most of them are clarified. However, since only one leg has been made up to the present, the bearing capability of the mechanism has not been verified; this problem is left for further work. On the other hand, other kinematic properties, i.e., dexterity and singularity, will be explored. In addition, we will apply the mechanism to a robot and assess its dynamic characteristics, gait planning, and control method, all of which are necessary aspects in practical robot applications and are worth investigating in future work.