Type Synthesis of a Novel 4‐Degrees‐of‐Freedom Parallel Bipedal Mechanism for Walking Robot

In this study, the conventional virtual chain method is refined and a 4‐degrees‐of‐freedom (DOF) parallel bipedal mechanism (PBM) for walking robots is presented. The proposed design has the advantages of low inertia, high load‐to‐weight ratio, and impact resistance. First, to solve the redundant motion‐transmission problem, a refined virtual chain method is proposed for the type synthesis of the 4‐DOF parallel thigh mechanism. Second, the relationship equation for each actuated wrench component in a reasonable actuated joint arrangement is derived, and the number and location of the actuated joints are determined. Third, by optimizing the design of the hip joint as a 2‐DOF counter‐centered five‐link spherical mechanism with different rotation centers, passive knee and ankle joints are designed based on the bionic principle. This design is performed to make the PBM isotropic and minimize inertia. The proposed PBM can withstand heavy loads and impacts owing to its humanoid crossed four‐link knee joint mechanism and compliant calf mechanism. Finally, the prototype is processed, and the rationality of the design and excellent performance of the PBM design are verified by dynamic and static simulations and gait experiments.


Introduction
Mobile robots are widely used in disaster rescue, space exploration, counterterrorism, patrols, and other applications.Robots significantly reduce the human workload while improving efficiency and human safety.5][6] Bipedal robots, a type of legged robot, have stronger mobility and environmental adaptability than other robots. [7]Therefore, bipedal robots have become an international research topic in recent years, and significant progress has been made through the joint efforts of several researchers.
In general, legged robots walk in realistic unstructured environments using legged mechanisms connected in serial, parallel, or hybrid. [8]The serial-legged mechanism is characterized by high speed, low cost, and a large motion space. [9,10]Tsagarakis et al. [11] designed a high-performance humanoid platform and built a seriallegged mechanism by developing a new serial elastic actuator that enabled a robot to move forward and backward, dynamically step on obstacles, and turn 180°in place.Ambrose et al. [12] built a serial-biped robot platform, AMBER-3M, and compared its energy efficiency.Galliker et al. [13] designed an online nonlinear model predictive controller based on the AMBER-3M platform, which enabled offline gait planning and online dynamic walking control.However, these serial-legged mechanisms have low load-carrying capacities and short actuator lifetimes.The hybrid-legged mechanism exhibits good compactness and has a large working space. [14,15]For example, Gim et al. [16] designed a 6-degrees-of-freedom (DOF) hybrid bipedal robot that achieved agile bipedal movement by performing foot-end workspace analysis and offline trajectory planning.Subsequently, the same team designed a large hybrid bipedal robot.Designing parallelly actuated hip joints and serially actuated knee joints results in a robot structure that is compact and lightweight, with low inertia. [17]owever, these hybrid-legged mechanisms are significantly complex for kinematic solving and control system construction.
For a bipedal robot to bear a relatively heavy load, the legged mechanism requires high stiffness and must realize real-time and accurate control.This will help accurately rectify the robot's posture in real time to prevent it from falling after interference.The legged mechanism is required to have the smallest possible cumulative error and a fast dynamic response.Furthermore, the robot is only able to achieve a basic humanoid gait during motion.[20] However, the parallel leg mechanism meets all the aforementioned requirements. [21,22]For example, BadriSproewitz et al. [23] were inspired by birds to design a parallel elastic-legged mechanism for the robot BirdBot, which can precisely realize self-engagement and disengagement.Wang et al. [24] designed a multimode amphibious bionic-legged robot equipped with a control system.Combining the defined boundary optimization algorithm with the analysis of robot kinematics resulted in a load-to-weight ratio of 4.23.However, most of the bipedal robots studied thus far are characterized by high inertia and low load-to-weight ratios. [25]Therefore, it is necessary to design a parallel bipedal mechanism (PBM) to provide an efficient walking capability to the robot.
The movement surface of human motion can be divided into three planes: sagittal, coronal, and horizontal.The three planes are perpendicular to each other and their intersection lines are called the sagittal, coronal, and vertical axes.While walking, human legs can achieve extension and flexion movements around the coronal axis in the sagittal plane, abduction and adduction movements around the sagittal axis in the coronal plane, and external and internal rotation movements around the vertical axis in the transverse plane (S). [26,27]Therefore, with reference to the human motion mechanism, for the PBM to achieve basic human gait during motion, the mechanism must have at least three DOFs.To increase motion flexibility but not the redundant DOFs, we set the number of target DOF to four.Therefore, we added 1-DOF for movement along the vertical axis (P).
In the present study, we developed a 4-DOF PBM, which can achieve circumferential rotation around the coronal, sagittal, and vertical axes as well as axial movement along the vertical axis.Additionally, the bipedal mechanism is characterized by low inertia and a high load-to-weight ratio, with a certain impact resistance.The remainder of this article is organized as follows: in Section 2, the advantages of the virtual chain method over other methods of parallel mechanism-type synthesis are presented.A redundant motion-transmission problem was observed for three-base twists with zero pitch.In Section 3, the differences between the refined and conventional virtual chain methods are presented, and the synthesis of three basic configurations of the SP-type parallel thigh mechanism (SP=PTM) using the refined method is described.In Section 4, the final configuration of the SP=PTM and a reasonable arrangement scheme for the actuated joints based on their selection criteria and screw theory are presented.In Section 5, the advantages of the isotropic mechanism are detailed, and the 2-DOF counter-centered five-link spherical mechanism for the hip joint is obtained by solving the kinematic equations.This rendered the PBM isotropic.
In Section 6, the motion mechanisms of the human knee and calf are introduced, and passive knee joint, ankle joint, and compliant calf mechanisms are designed.In Section 7, the performance of the designed PBM is verified by simulation and experiment.Finally, in Section 8, the conclusions of this study are presented.

Problem of Redundant Motion-Transmission Based on Conventional Virtual Chain Method
In recent years, researchers have proposed several systematic methods for synthesizing parallel mechanisms with good performances.39][40] Except for the virtual chain method, these synthesis methods have complicated derivations, are not universally applicable, and can only synthesize partial configurations.
The virtual chain method is a suitable solution to these problems. [41,42]In brief, a virtual chain is used to describe the motion pattern of a prismatic platform with a parallel mechanism in serial or parallel chains.For a given motion pattern, the configuration of the virtual chain is not unique to the same wrench system.Typically, we choose the simplest kinematic chain that can express the motion pattern well.The conventional virtual chain method transforms the type synthesis of a parallelmechanism leg kinematic chain (LKC) into that of a single-loop kinematic chain.It comprises the same virtual chain corresponding to the kinematic mode of the prismatic platform and different LKCs.The obtained LKCs are then assembled in different combinations to obtain a variety of parallel mechanisms that satisfy the requirements. [43,44]or an SP-type parallel mechanism containing n LKCs, if a set of the base twist of the twist system contains three twists with zero-pitch $ 01 , $ 02 , and $ 03 and if the configuration is synthesized according to the conventional virtual chain method, any LKC of the resulting parallel mechanism contains at least twotransmission twists [41] that intersect at one point.This is such that this parallel mechanism will have at least four-transmission twists that intersect at one point, as shown in Figure 1.To generalize this parallel mechanism, C n is used to denote the nth LKC.The dotted line indicates the possible joints and the broken line indicates the joints and links omitted in each LKC.
Figure 1 shows at least four-transmission twists intersecting at point Q: $ T1 and $ T2 of C 1 and $ T3 and $ T4 of C 2 .We may consider intersection point Q as the coordinate origin, and any transmission twist $ Tn intersecting at Q point in this parallel mechanism can be expressed as where L, M, and N are real numbers that are not zero simultaneously.
The dimensions of the spinor system of the transmission motion are expressed as where $ T0i denotes the ith base-transmission twist with zero pitch.
The total dimensions of the line geometry space are constant and combined with the basic criterion of the Grassmann line geometry. [45]Thus, we could obtain the spatial line graph of the transmission twist intersecting at a point in this parallel mechanism and the isodimensional equivalent spatial line graph, as shown in Figure 2 and 3.
According to the analysis mentioned before, when more than three-transmission twists intersect at one point in a parallel mechanism, at most, three twists are independent of each other.The existence of such redundant transmission twists reduces the motion-transmission efficiency of the parallel mechanism and increases its complexity.This inconveniences the subsequent motion control in each joint.

Decomposition of Virtual Chain of SP-Type Parallel Thigh Mechanism
To solve the redundant motion-transmission problem of the SP-type parallel mechanism, we refined the conventional virtual chain method.The differences between the refined and conventional methods include the following: i) three different subvirtual chains are used to jointly represent the motion patterns of the moving platform and ii) each LKC in the parallel mechanism is derived from the corresponding different subvirtual chains.
We designed a PBM thigh using the SP-type parallel mechanism.The twist system was composed of three $ 0 base twists intersecting at a point of the base with zero pitch, and one $ ∞ base twist with an infinite pitch for the general configuration of SP=PTM.The instantaneous powers of the twist and wrench of SP=PTM were zero and the wrench system of the prismatic platform was a 2 À $ r 0 system.The SP-type virtual chain that can provide two constraint forces and connect the prismatic platform, as well as the base, is shown in Figure 4.The axes of all $ r 0 pass through the rotation center of the S joint and are perpendicular to the axis of the P joint.
The redundant motion transmission of SP=PTM contains only SP in the virtual chain (V=SP) described in Section 2. Thus, we used different subvirtual chains V i to jointly represent the motion pattern of the prismatic platform.These subvirtual chains must satisfy the following relationship Equation ( 3) indicates that there are three subvirtual chains of V=SP: V 1 =SP, V 2 =SPP, and V 3 =SPPP, as shown in Figure 5.In the following section, the synthesis of the LKC for each of the three subvirtual chains is described.To enable the prismatic platform to perform a 4-DOF SP-type motion, it is necessary to ensure that each subvirtual chain can form a single-loop kinematic chain, with each LKC having the same DOF as each subvirtual chain.

Decomposition of Wrench System of SP-Type Parallel Thigh Mechanism
The decomposition of the wrench system of the SP=PTM includes determining all wrench systems of LKCs, all their combinations, and the number of overconstraints for each combination.The linear combination of the base wrench of the moving platform system constituted each leg wrench system (LWS) in the general configuration of the SP=PTM.Therefore, LWS was either a 2 À $ r 0 system, 1 À $ r 0 system, or 0 À $ r 0 system.Equation (3) shows that there are at most three LKCs; thus, the overconstraint σ of the SP=PTM containing n LKCs can be expressed as where c denotes the dimensions of the wrench system for the SP=PTM, and c i denotes the dimensions of the wrench system for the ith LKC.Without considering redundant DOFs, the DOF F p of SP=PTM was equivalent to the dimension C of the twist system.c ¼ 6 À C was incorporated into Equation (4) to obtain the expression for the dimensions of the LWS LKC combinations containing different numbers of overconstraints were then obtained by selecting different values for each parameter in Equation ( 5) in the optional range, as shown in Table 1.

Type Synthesis of LKCs
If the type synthesis of the LKC is performed using the conventional virtual chain method, it is necessary to form a 4-DOF single-loop kinematic chain by combining any LKC with the SP-type virtual chain.Subsequently, the geometric conditions that the leg twist system (LTS) and LWS should satisfy were obtained for each of the three wrench systems.However, this method results in under-constrained single-loop kinematic chains.This increases the number of singular configurations of the finally obtained LKC and, thus, some uncontrollable DOFs.To solve this problem, we performed the type synthesis of the 4-DOF single-loop kinematic chain containing the 2 À $ r 0 wrench system and the SP-type subvirtual chain; the 5-DOF single-loop kinematic chain containing the 1 À $ r 0 wrench system and the SPP-type subvirtual chain; and the 6-DOFs singleloop kinematic chain containing the 0 À $ r 0 wrench system and the SPPP-type subvirtual chain.
After combining a subvirtual chain with an LKC to form a single-loop kinematic chain, the relative DOF at the beginning and end of the LKC were limited by the subvirtual chain.Thus, the DOFs F s of this single-loop kinematic chain were equivalent to those of the subvirtual chain it contained.This can be expressed as where f denotes the number of joints in a single-loop kinematic chain.c ¼ 6 À C is substituted into formula (6) to obtain the number of joints contained in the single-loop kinematic chain, which can be expressed as follows The joints involved in this study included two types of simple kinematic joints: revolute joint (R) and prismatic joint (P).A spherical composite joint is composed of three revolute joints, whose axes pass through the center of rotation.It comprises multiple revolute and prismatic joints.The 2D surface of the torus composite joint consists of two revolute joints, whose axes are perpendicular to each other but do not intersect.Table 1.Combinations of leg-kinematic chains with different numbers of overconstraints.

Geometric Conditions to Be Satisfied by LTS and LWS in a
Single-Loop Kinematic Chain Containing 2 À $ r 0 Wrench Systems and SP-Type Subvirtual Chains Equation (7) shows that the 4-DOF single-loop kinematic chain containing the 2 À $ r 0 wrench system, and SP subvirtual chain had eight joints, implying that each LKC had four joints.The linear dependence of the wrench system is unaffected by the choice of the coordinate system.Thus, for convenience, we considered the intersection of two base wrenches $ r 01 and $ r 02 of the LWS as the origin of the coordinate system.Subsequently, the set of base wrenches in the LWS can be expressed as follows where The reciprocal product of any set of basis twists $ j and basis wrench $ r 0i should satisfy where Δ is an antisymmetric unit matrix that can be expressed as By solving Equation ( 9), a set of basis twists of LTS can be expressed as follows By linearly combining the aforementioned four base twists, a general expression for the twist of this single-loop kinematic chain can be obtained as follows where a, b, c, and d are arbitrary constants that cannot be zero simultaneously.When the conditions s T s 0 ¼ 0 and s T s ¼ 1 are satisfied, Equation (12) degenerates to a zero-pitch unit twist. where Equation ( 15) denotes a revolute joint.Equation ( 8) and (13) show that s 0T s r i 6 ¼ 0 and that a and b are arbitrary constants.Thus, all revolute joint axes intersected at a point with the two base wrenches.Equation (11) shows that the set of the base twist of LTS contains three zero-pitch twists, $ 1 , $ 2 , and $ 3 as well as one twist $ 4 with infinite pitch.To match the motion of the SP subvirtual chain, the LKC must contain a spherical composite joint and a prismatic joint.Considering that there are no redundant DOFs in this single-loop kinematic chain, the center of rotation of the spherical composite joint in the LKC coincided with that of the S joint in the subvirtual chain.Additionally, the axes of the prismatic joints in the LKC and the subvirtual chain were parallel to each other.
Let a ¼ 0, b ¼ 0, and c ¼ 0 and let the factors be regularized.Equation ( 12) degenerates into a unit twist with an infinite pitch as follows: where 13) denotes a prismatic joint.In combination with Equation ( 8), we derived $ b∞ Δ$ r 0i ¼ 0. This implies that both prismatic joint axes were perpendicular to the base wrench.In summary, a 4-DOF single-loop kinematic chain containing the 2 À $ r 0 wrench system and the SP subvirtual chain are represented as shown in Figure 6.
Removing the SP subvirtual chain from the aforementioned kinematic chain yielded the following LKC: posed of X joints, whose center of rotation coincided with the center of rotation of the S joints in the subvirtual chain (X is a revolute or prismatic joint).X denotes that the axes of all X joints intersect at the same point.X • • • X ð Þ F denotes a planar composite joint composed of X joints in the same plane as the prismatic joints in the subvirtual chain.

Geometric Conditions to Be Satisfied by LTS and LWS in a
Single-Loop Kinematic Chain Containing 1 À $ r 0 -Type Wrench Systems and SPP-Type Subvirtual Chains Equation (7) shows that the 5-DOF single-loop kinematic chain containing 1 À $ r 0 wrench system and SPP subvirtual chain contains 10 joints.Each LKC comprised five joints.The linear dependence of the wrench system is independent of the choice of coordinate system.For convenience, we chose the origin of the coordinate system to be on the base wrench $ r 01 of LWS.Subsequently, the set of base wrenches of the LWS can be expressed as where 9), a set of basis twists of LTS can be expressed as follows: By linearly combining the five base twists mentioned before, a general expression for the twist of this single-loop kinematic chain is obtained as When the conditions s T s 0 ¼ 0 and s T s ¼ 1 are satisfied, Equation (18) degenerates to a zero-pitch unit twist. where Equation ( 19) represents a revolute joint.Equation ( 16) and (19) show that s 0T s r 6 ¼ 0, together with the fact that a, b, and e are arbitrary constants.Thus, this base constraint rotational quantity intersected or was parallel to all the rotational joint axes.Equation (17) shows that the set of base twists of LTS contains three zero-pitch twists $ 1 , $ 2 , and $ 3 , and two twists $ 4 and $ 5 with an infinite pitch.To match the motion of the SPP subvirtual chain, the LKC must contain a spherical compound joint, a planar composite joint, or a 2D surface of the torus composite joint.Considering that there are no redundant DOFs in this singleloop kinematic chain, the center of rotation of the spherical compound joint in the LKC coincided with the center of rotation of the S joints in the subvirtual chain, and the plane of the motion trajectory for the planar composite joint in the LKC was parallel to the plane of the motion trajectory for the two prismatic joints in the subvirtual chain.The common perpendicular to the axes of the two rotating joints in the 2D surface of the torus composite joint was parallel to the normal of the plane of the motion trajectory for the two prismatic joints in the subvirtual chain.
Let a ¼ 0, b ¼ 0, and c ¼ 0 and let the factors be regularized.Equation (12) degenerates into a unit twist with an infinite pitch as follows: where Equation ( 20) denotes a prismatic joint; in combination with Equation ( 16), we derived $ b∞ Δ$ r 0i ¼ 0, which implies that the base wrenches were perpendicular to all the axes of the prismatic joint.In summary, a 5-DOF single-loop kinematic chain containing the 1 À $ r 0 wrench system and SPP subvirtual chain is represented as shown in Figure 7.
Removing the SPP subvirtual chain from the aforementioned kinematic chain yields the following LKC.
. X denotes that the axes of all X joints are par- torus composite joint consists of multiple X-joints, and X ↔ denotes that the axes of all X joints are perpendicular to each other.

Geometric Conditions to Be Satisfied by LTS and LWS in a
Single-Loop Kinematic Chain Containing 0 À $ r 0 -Type Wrench Systems and SPPP-Type Subvirtual Chains Equation (7) shows that the 6-DOF single-loop kinematic chain containing 0 À $ r 0 wrench system and SP subvirtual chain contains 12 joints.Furthermore, each LKC contained six joints.The set of base twists of the LTS contained three twists with zero pitch and three twists with an infinite pitch.To match the motion of the SPPP subvirtual chain, the LKC must contain spherical and spatial composite joints.In addition, there were no redundant DOFs in this single-loop kinematic chain.The centers of rotation of the spherical composite joints in the LKC coincided with the centers of rotation of the S joints of the subvirtual chain.If the twists corresponding to the revolute and prismatic joints in the spatial composite joint are linearly independent, the axes of the joints in the LKC can be along any direction.In summary, the 6-DOF single-loop kinematic chain containing 0 À $ r 0 wrench system and SPPP subvirtual chain is represented as shown in Figure 8.
Removing the SP subvirtual chain from the aforementioned kinematic chain yields the following LKC: denotes the spatial composite joint that can perform the spatial motion composed of X joint.

Assembly and Selection of LKCs
Considering the lightweight requirements, the number of LKCs should be minimized.Therefore, we considered only two types of LKCs.The LKCs obtained from the type synthesis were assembled to obtain the SP=PTM for various configurations, as listed in Table 2.Only a few LKC types are listed in the table, and other similar LKC types can be obtained by exchanging the positions of the moving and rotating joints.
First, to ensure that the configuration has a continuous workspace for a given initial position and avoid constraint singularity, the two selected LKCs should ensure that the wrench system does not reduce the order.Second, to avoid an outputtransmission singularity, the two selected LKCs should ensure that the transmission wrench systems do not reduce the order.Finally, to ensure that the linear combination of the wrench systems of the two selected LKCs can form a 2 À $ r wrench system, the normal of the equivalent planes formed by the axes of each rotating joint should not be parallel to the same plane.In addition, if prismatic joints are selected as the actuated joints, the performance requirements of the actuators and guide rail are high, and the passive prismatic joints deteriorate the motion performance of the SP=PTM.Therefore, all LKCs containing moving joints were discarded in the proposed design in this study.The SP=PTM types selected are described as

Selection of Actuated Joints
The arrangement of actuated joints significantly affects the performance of the mechanism.Additionally, to minimize the inertia of the SP=PTM during motion and optimize the motion performance, the actuated joints should be evenly distributed on the two LKCs and as close as possible to the base. [46]Let $ ri ∈ =j denotes the actuation wrench provided by the jth actuated joint of the ith LKC acting on the moving platform.This actuation wrench was reciprocal to the transmission twist corresponding to all joints except the jth joint within the ith LKC (i ¼ 1, 2j ¼ 1, 2, : : : , f ).When the actuated joints are locked, to ensure that the SP=PTM with 4 DOFs has 0 DOF in a general configuration, four actuated joints should be selected to generate four actuation wrenches.To completely restrict the motion of the SP=PTM, we obtained 8 > > > < > > > : where $ ri a∈ =j is a 2D column vector and $ ri b∈ =j is a 4D column vector.Considering that the linear dependence of the wrench system is independent of the choice of the coordinate system, for convenience, we considered the intersection of the two base wrenches $ r 01 and $ r 02 of the LWS as the origin of the coordinate system.Additionally, given that the z-axis is perpendicular to the wrench system of the moving platform, we obtained where i is a 2D unit vector along the x-axis, j is a 2D unit vector along the y-axis, and 0 is a 2D zero vector.By incorporating Equation (22) into Equation ( 21), we obtained SP=PTM contained a set of mutually parallel rotating joints and the other LKC contained a set of mutually perpendicular rotating joints.The proposed SP=PTM can satisfy the condition that there exists a set of mutually uncorrelated components of the actuation wrench $ ri b∈ =j in either configuration, whereas other types of SP=PTM cannot satisfy the condition of (21).A schematic of the SP=PTM that satisfies the requirements is shown in Figure 9.
As shown in Figure 9 R ↔ T type of SP=PTM, the first wrench passed through the center of rotation S 1 of the spherical composite joint of the first LKC and was parallel to the R joint axis in the planar composite joint.Additionally, the second wrench passed through the center of rotation S 2 of the spherical composite joint of the second LKC and coincided with the common vertical line of the two R ↔ joint axes in the planar composite joint.The actuated wrench of the two actuated joints of the first LKC is the intersection of the plane in which the axes of the two R joints within the planar composite joint are located as well as the plane in which the axes of the two non-native joints within the spherical composite joint are located.The actuated wrench of the two actuated joints of the second LKC is in the plane where the axes of the two non-native joints in the spherical composite joint are located and parallel to the axis of one of R ↔ joints in the planar compound joints and perpendicular to and intersecting with the axis of the other R ↔ joint.Therefore, we selected the four R-joints closest to the base as actuated joints.

Optimal Design of Hip Joint Configuration
The isotropy of a mechanism is a property in which the condition number of the kinematic Jacobian matrix of the mechanism reaches a minimum value per unit.Thus, the joint velocity can be solved for each joint based on a given series of Cartesian velocities for the end effector of the mechanism.This implies the same for the end-effector output force in Cartesian space based on a given series of joint moments for the end effector. [47][50] Therefore, it is necessary to isotropically design a mechanism to optimize its motion performance.Figure 10 shows a schematic of the PTM mechanism with two LKCs.
For ease of description, we introduced the following notation: i ¼ 1, 2, A i $ E i denotes the five joint axes in the ith LKC. a i $ e i denotes the unit direction vectors corresponding to the five joint Table 2. Different combinations of LKCs that can constitute SP = PTM.tion for the ith LKC can be obtained according to the geometric relationship of each joint axis as shown in Figure 10 as. [51]i θ where J i is the 5 Â 5 Jacobian matrix of the ith LKC, which can be expressed as follows: The D 1 and E 1 axes remained parallel to each other in any configuration, whereas the D 2 and E 2 axes were perpendicular to each other in any configuration.Thus, the Jacobian matrices of the two LKCs can be expressed separately according to Equation ( 25) as follows: To produce the ideal motion of a moving platform, we only needed to control the motion of the two actuated joints of each LKC. [51]To eliminate the influence of the remaining three nonactuated joints in each LKC, considering the geometric relationship of the SP=PTM joint axes, we considered a 5 Â 2 matrix I i , which is expressed as where the vectors represented by each column of the matrix I i and those represented by the last three columns of the Jacobi matrix J i are orthogonal to each other.We multiplied both ends of Equation ( 24) with I T i and combined them with Equation ( 27) to obtain the kinematic equation for the SP=PTM.where D is the positive Jacobi matrix and K is the inverse Jacobi matrix, [52] which is denoted as follows For the SP=PTM to be isotropic, D and K must satisfy the following conditions in at least one configuration: where 0 is a 2 Â 2 zero matrix, 1 is a 2 Â 2 unit matrix, and α and β are real constant numbers.
According to Equation (30,33,34), combined with the geometric relationship of the joints of the SP=PTM derived from the previous type of synthesis.The SP=PTM satisfies the isotropic condition only when a i ¼ b i .This requires that the axes of the two actuated joints of each LKC coincide.Therefore, to avoid the singularity of the positive Jacobi matrix, we used a 2-DOF counter-centered five-link spherical mechanism instead of a spherical composite joint, and the final two LKCs obtained for the SP=PTM are shown in Figure 11.
As shown in Figure 11, the five joint axes of the 2-DOF countercentered five-link spherical hip joint of the ith LKC intersected at point S i .To make the 2-DOF counter-centered five-link spherical hip joint equivalent to the original spherical compound joint, the unit direction vectors r oia and r oib of the line through point O i with the two actuated joint axes must be orthogonal to the unit direction vectors a i1 and b i1 of the adjacent joint axes of the corresponding actuated joints, respectively.The nonsingular positive Jacobi matrices of the isotropic SP=PTM are expressed as follows: In addition, by designing the hip joints as 2-DOF countercentered five-link mechanisms with different centers of rotation, the PBM was made isotropic and all actuated joints were fixed to the base.This further reduces the inertia of the robot during motion and improves walking efficiency to a certain extent.

Design of Humanoid Knee and Compliant Calf Mechanism
Currently, most legged robots have one active DOF each at the knee and ankle joints (except for the point foot).The benefits of such a design include not only a simple hinge structure and easy processing but also the convenience of motion control.However, similar to the 3-DOF compliant leg in previous studies, [53] the inclusion of actuator components such as motors, servos, and hydraulic actuators in the lower limbs increases the mass of the robot as well as the inertia during the robot's motion.Even if it is arranged as upward as possible in the lower limbs, similar to the arrangement of ankle joint actuator components of the robot Cassie, [53] this will affect the walking efficiency to a certain extent.The point-legged mechanism without ankle DOFs, similar to the robot ANYmal, can satisfy the motion demand of the quadruped robot [54] but not the stability demand of the standing phase of the bipedal mechanism.In addition, the impact force generated by the contact between the footplate and the ground makes the motion control uncertain and may damage the actuator parts.Therefore, it is necessary to design a cushioning mechanism that can absorb the impact force.To make the PBM lighter, reduce the inertia of its motion, minimize the impact of the ground reaction force on the robot's motion, and achieve a stationary stand similar to that of a human, it is necessary to design high-strength passive knee joints, compliant calf mechanisms, and passive ankle joints.The structural design of the knee joint directly affects the motion performance of the robot, [55] and a reasonable knee joint structure can ensure the stability of the support phase of the bipedal mechanism and flexibility of the swing phase.Currently, the knee joints of most legged robots and lower-limb exoskeletons use single-axis hinges. [56,57]Although this hinge is straightforward and easy to manufacture, it has low strength and a limited rotation angle owing to the other structural components of the robot. [58]The human knee joint is the most weightbearing and complex joint among all joints, and previous studies found that the movement pattern of the knee joint is not a simple flexion-extension movement but a complex multi-DOF movement pattern with flexion-extension, rolling, sliding lateral shift, and axial rotation. [59]The anterior cruciate ligament (ACL) and the posterior cruciate ligament (PCL) between the femur and the tibia are the most important and strongest structures to maintain the stability of the knee joint, and the instantaneous center of rotation of the knee joint becomes variable and the trajectory resembles a "J" curve owing to the presence of these ligaments. [60]This increases the distance of the foot from the ground during movement and reduces the energy expenditure. [61]The Achilles tendon is the thickest tendon in the body; it begins below the triceps muscle of the lower leg and ends at the radicular tuberosity.Because the Achilles tendon is composed of mechanically sensitive tendon cells, it can temporarily store and release large amounts of energy during human walking, thereby reducing the impact of the ground and increasing walking efficiency. [62]Inspired by the structure of the human lower-limb skeletal muscles, especially the knee joint and Achilles tendon, we designed the knee joint using a crossed four-link mechanism with a variable instantaneous center of rotation and used a spring accumulator to simulate the Achilles tendon, as shown in Figure 12.The ACL linkage in the knee joint is connected to the posterior femoral linkage at the upper end and the tibial linkage at the lower end.The PCL is connected to the anterior femoral linkage at the upper end and the triceps linkage at the lower end.The ends of the spring accumulator are connected to the triceps and Achilles tendon linkages.
When all the actuated joints are locked, preventing the PBM rods from moving relative to each other under external forces is necessary to maintain the stability of the overall configuration of the mechanism.The mechanism comprising the knee, calf, and ankle must ensure that there is only 1 DOF at the ankle, which is provided by the spring accumulator.The DOFs of the mechanism shown in Figure 12 can be determined according to the following equation: where v, p l , and p h denote the number of movable linkages (v ¼ 5), the number of low pairs (p l ¼ 6), and the number of high pairs (p h ¼ 0), respectively.Therefore, considering the lightweight requirement, a composite hinge should also exist while maintaining the number of bars from increasing, as shown in Figure 13 (the shaded lines in the figure indicate linkages with the configuration completely fixed when the actuated joint is locked).
Figure 14 shows the three basic planes (sagittal, coronal, and horizontal) and the three basic axes (sagittal, coronal, and vertical) of the PBM during motion.strength.First, the output torque of the driving joint of the servo is considered.After performance comparison, the HBL388-type servo produced by MARK STAR Company is selected as the actuator of the four driving joints.The weight of the servo is only 93.3 g, and the rated output torque can reach 6.8 Nm.We use the dynamics simulation software Adams to perform the dynamics analysis.First, the foot of the PBM is fixed, and a force with an amplitude increasing from 0 to 225 N and a vertical downward direction is applied to the top of the PBM, so as to simulate the maximum load that the robot can bear when standing on the ground.Then, corresponding constraints are applied to each joint of PBM.The driving functions of the four driving joints are set to zero.To converge the solution results, the simulation step is set to 0.01 and the total duration is 5 s.The simulation model and simulation results are shown in Figure 15.

Verification of the Simulation and Experiment
As shown in Figure 15, with an increase in the load, the output torque of the four driving joints also increased.When the PBM was subjected to 225 N of static load, the output torques of the four driving joints were approximately 4.9, 6.1, 4.6, and 2.5 Nm successively, all of which did not exceed the torque capacity of the selected servo.Therefore, the PBM can withstand a static load of more than 45 kg, when standing on both feet.Considering the lightweight design, all parts of the robot were made of carbon fiber tubes and aluminum alloy and its weight was less than 2.5 kg, so the designed biped robot had a load-to-weight ratio of more than 18.
In terms of structural strength, we used the finite-element simulation software ANSYS to evaluate the structural strength of the designed PBM.After the corresponding carbon fiber and aluminum alloy materials are used to synthesize each component, as the PBM contains both shell and entity, the two elements were divided into grids with triprism and hexahedral elements, respectively.To ensure the quality of the grid and make sure that the units at the contact point between the shell and the entity do not overlap and, thus, produce constraints, the size of the unit was set to 4 mm.The resolution was set to 2, and the overexpansion ratio and maximum number of layers were set to 0.272 and 5, respectively, and solved using a mechanical solver.A vertical downward force of 225 N was applied to the top of the bipedal mechanism whose end was fixed.The von Mises stress distribution is shown in Figure 16.
As shown in Figure 16, when the PBM was subjected to a load of 22.5 kg, the maximum stress on the carbon fiber tube was 186.05 MPa, and the yield strength of the selected carbon fiber   tube was 600 MPa, far from reaching its yield strength.The yield strength of the aluminum alloy bar was 280 MPa.According to the previous analysis, the output torque of driving joint 2 was the largest.Therefore, we only needed to consider the stress of the connecting rod on the side of driving joint 2, whose maximum stress was 74.402 MPa, which also failed to reach its yield strength.Therefore, when the robot was subjected to a static load of 45 kg, the designed PBM satisfied the strength requirements.Therefore, it is again verified that the load-to-weight ratio of the designed PBM can exceed 18.

Verification of the Impact Resistance
To verify the ultimate impact force that the designed PBM could withstand, we used the dynamic simulation software Adams to perform a dynamic analysis.To simulate the movement of the robot on a general road surface, we referred to the gait characteristics of human movement, that is, when the gait switches, the heel lands first.Therefore, the top end of the PBM was fixed and vertical pulse forces with different amplitudes were applied to the rear position of the robot foot to simulate the impact force from the ground during the movement of the robot.The amplitude of the pulse force randomly fluctuated between 50 and 125 N, and the output torque changes in each driving joint are shown in Figure 17.
As shown in Figure 17, when the robot was subjected to an impact force with an amplitude of 125 N, the output torque of driving joint 2 was the largest, with a size of 6.2 Nm, and did not exceed the torque capacity of the servo.This indicates that the designed PBM can withstand an impact force of five times its gravity.Furthermore, considering the reaction force of feet from the ground during walking is about 1-1.5 times of the weight of human beings, and the reaction force of feet from the ground during running is about 2-2.5 times of the weight of human beings.Therefore, the PBM designed in this study satisfies the requirements for bearing the impact force during movement.

Verification of the Gait and Workspace
To further counteract the impact force from the ground during movement and thus extend the service life of the servo, the PBM's feet consist of a carbon fiber plate on the top half and EVA foam on the bottom half.The height of the prototype is 60 mm.To verify whether the designed PBM can achieve basic gait and whether the working space at the foot can meet the basic gait requirements, we conducted a comparison experiment between robot walking and human walking.The experimental equipment included computer, power supply, target ball, and optical tracking system, as shown in Figure 18.
The specific experimental steps include: firstly, the target ball is fixed on the foot of the human and PBM, respectively, and then the motion trajectory of the foot of the human and PBM is captured by the optical tracking system.Finally, the captured motion As shown in Figure 19, the bipedal mechanism designed by us can realize basic 4-DOF gait, including flexion and extension movement around the coronal axis in the sagittal plane, adduction and abduction movement around the sagittal axis in the coronal plane, internal and external rotation movement around the vertical axis in the horizontal plane, and standing in place movement along the vertical axis.There is no interference of each bar during the movement, so the designed PBM can complete the basic human-like gait.Moreover, As shown in Figure 20, since it is impossible for people in the walking process to achieve the same step length and step height in each step, but the robot can achieve it through program control, so the surface of the working space of the human foot has irregular protrudations, which does not look like the regular surface shape of the working space of the PBM foot.But in general, the limiting displacements of the robot's foot in the x axis, y axis, and z axis all exceed the limiting displacements of the human's foot in these three directions, and they have the same contour line arrangement order.Therefore, the workspace surface of the robot can completely envelope that of the human.In other words, the full working space can also meet the basic gait requirements, thus verifying the rationality of PBM design.

Comparison with Several Other Representative Biped Robots
A comparison of the biped robot designed in this study with several representative biped robots in terms of size(high), weight, load, mechanism strength (special configuration), movement inertia (actuator position), and impact resistance (anti-impact mechanism) was performed, and the results are presented in the Table 3.
As shown in the Table 3, first, in terms of size, the PBM designed in this study was the smallest, meaning that it can perform tasks more flexibly in smaller spaces.In terms of mechanism strength, PBM has a crossed four-link knee joint, which is strong enough to withstand higher loads compared to six-link mechanisms and double-closed loop mechanisms.In terms of weight and load, PBM has the lightest weight.Although Atlas [63] can withstand a load of up to 680 N, it is at the expense of increasing its own weight.According to the load ratio of PBM designed in this article, to achieve the same load requirements as atlas, only the weight ratio of PBM needs to be increased to 3.78 kg (provided that there is a servo that can provide sufficient output torque) which is much less than 75 kg.In terms of movement inertia, the servo of the designed PBM is completely horizontally arranged in the pelvis, which means that the servo does not swing with the leg swing during the movement of the robot.Compared with the actuator mounted on the leg like Cassie [64] and Atlas robots, the movement inertia is smaller.Finally, in terms of impact resistance, Leonardo [65] does not use a special mechanism to resist the impact force, but the PBM designed in this article can withstand the impact force of more than five times its own weight due to the design of humanoid compliant legs.In conclusion, the PBM designed in this article has significant advantages in the comprehensive performance of load-toweight ratio, mechanism strength, movement inertia, and impact resistance.A more intuitive comparison is shown in Figure 21.

Conclusion
In this study, a 4-DOF PBM for walking robots was designed, which has the advantages of low movement inertia, high load-to-weight ratio, and certain impact resistance.The achievements of this study are reflected in the following three aspects: the improved virtual chain method was used to synthesize the configuration of the parallel thigh mechanism, optimize the hip joint, and design the lower leg mechanism.The performance of the designed biped mechanism was verified by simulation and experiment.First, based on the line geometry and spinor theories, the redundant motion transfer problem existing in the configuration synthesis of a parallel mechanism containing three fundamental motion spinors with zero pitch using the traditional virtual chain method was pointed out.Therefore, an improved virtual chain method was proposed.The differences between the improved and conventional virtual chain methods are as follows: i) three different subvirtual chains are used to jointly represent the motion patterns of the moving platform and ii) each LKC in the parallel mechanism is derived from the corresponding subvirtual chains.The main steps in the refined virtual chain method included the decomposition of the SP=PTM virtual chain, decomposition of the wrench system of the SP=PTM, type synthesis of LKCs, and assembly and selection LKCs.All three basic configurations of the SP=PTM that satisfied the requirements were synthesized on the refined virtual chain method.
Subsequently, a unique SP=PTM configuration that satisfies the relation condition was obtained.The four R joints closest to the base were selected as the driving joints such that the PBM had a small movement inertia.By solving the kinematics equation and according to the conditions that the isotropic mechanism should be satisfied, the hip joint was used as a 2-DOF counter-centered five-link spherical mechanism with different rotation centers to make the PBM isotropic.The nonsingular positive Jacobi matrices of the two LKCs were provided.In addition, the hip joint was designed such that all actuated motors were fixed to the base, thus further reducing inertia and improving motion efficiency.The passive knee and ankle joints were designed based on the motion mechanism of the human lower limb.The knee joint of a crossed four-link mechanism with a variable instantaneous rotation center that can withstand a high load was designed.An ankle joint with 1-DOF was designed to ensure the stability of the overall configuration.A humanoid-compliant calf mechanism was designed to resist ground impact during motion.
Finally, dynamic and static simulation analyses of the PBM were conducted using Ansys and Adams, respectively.The analysis results showed that the PBM can withstand a load of 45 kg and a load-to-weight ratio of more than 18.Furthermore, the bipedal mechanism can withstand an impact force of more than five times its weight.The prototype was made of carbon fiber, aluminum alloy, and EVA foam as raw materials to satisfy the requirements of high strength and low weight.The prototype was 320 mm long, 150 mm wide, and 596 mm high and weighed 2.45 kg.Subsequently, a dynamic walking experiment was conducted to capture the foot movement trajectory and compare it with a human foot trajectory.The experimental results showed that the designed PBM can realize a basic human-like 4-DOF gait and stationary standing, and not all the links interfered in the process of movement, which verified the rationality of the design.Compared with the most representative biped robots, we observed that the biped robot designed in this study has significant advantages in the comprehensive performance of load-toweight ratio and movement inertia, thus having a great potential in disaster relief such as transporting materials and rescuing the injured in the disaster area after an earthquake.Our future research will focus on integrating onboard power and controllers into biped robots as well as exploring practical applications in disaster relief and post-disaster reconstruction.

Figure 1 .
Figure 1.Parallel mechanism with three-base twist of zero pitch.

Figure 2 .
Figure 2. Spatial line graph of transmission twist system.

Figure 7 . A 5 -
Figure 7.A 5-DOFs single-loop kinematic chain containing 1 À $ r 0 LWS and SPP subvirtual chains.a) The first, b) the second, c) the third, d) the fourth, e) the fifth, f ) the sixth, and g) the seventh kind of single-loop kinematic chain.

Figure 8 .
Figure 8.A 4-DOF single-loop kinematic chain containing 0 À $ r 0 LWS and SPPP subvirtual chains.a) The first, b) the second, c) the third, and d) the fourth kind of single-loop kinematic chain.

:
RÞ G ðPÞ F axes in the ith LKC; and f denotes the unit direction vector that coincides with the common vertical line of the D i and E i axes.θ ai $ θ ei denotes the angular displacements of the five joints in the ith LKC.s bd denotes the distance between B i and D i axes.s de denotes the distance between D i and E i axes.A i , B i , and C i axes intersected at the spherical composite joint rotation center S i point.For convenience, point O i , which is the intersection of C i and D i axes, was considered the operating point of the ith LKC; point P is the operating point of the moving platform and r i denotes the direction vector of the operating point O i pointing toward the operating point P.The 5D joint velocity vector of the ith LKC can be expressed as θ The twist of operation point of the ith LKC can be expressed as t i ¼ ω T , o : T i Â Ã T , and the kinematic rela-

Figure 10 .
Figure 10.Schematic diagram of the mechanism of two leg kinematic chains (LKCs) of PTM.a) The original LKC in the back and b) the original LKC in the front.

Figure 11 .
Figure 11.Schematic diagram of the mechanism of two isotropic LKCs of PTM.a) The optimized LKC in the back and b) the optimized LKC in the front.

7. 1 .Figure 12 .
Figure 12.Schematic diagram of lower limb based on humanoid principle.a) Schematic diagram of basic structure of human knee joint and calf, b) schematic diagram of the mechanism including knee joint and calf.

Figure 13 .
Figure 13.Schematic diagram of the passive ankle joint.

Figure 15 .
Figure 15.Check of the output torque of biped mechanism.a) Constrained configuration of bipedal mechanism and b) diagram of output torque variation with load.

Figure 16 .
Figure 16.Strength check of bipedal mechanism.a) The stress on the whole bipedal mechanism, b) the stress of the most stressed aluminum alloy bar, c) the stress of the second most stressed aluminum alloy bar, and d) the stress of the most stressed carbon fiber bar.

Figure 17 .
Figure 17.Check of the output torque of biped mechanism.a) The impact force on the PBM and b) the output torque of each driving joint.

Figure 19 .
Figure 19.Basic gait of bipedal robot with 4-DOF.a) Flexion and extension, b) adduction and abduction, c) internal rotation and external rotation, d) stand up and e) marking time.

Figure 20 .
Figure 20.Comparison of PBM and human workspace.a) Workspace surface of PBM and b) workspace surface of human.

Table 3 .
Comparison with other biped robots.