2.1. Conceptual design I

The kinematic structure and a CAD model of the first conceptual design are given in Fig. 1. The kinematic structure is (UPU-2Pa)-(UPU-2Pa) where U stands for universal joint, P stands for prismatic joint, 2Pa stands for two parallel loops are serially attached to each other. Underlined joints are actuated, and Pa represents a parallelogram loop. This mechanism consists of a hip platform, a middle platform (knee), and a moving platform (foot), four rigid links and two parallelograms. The motion of the middle and moving platforms is analyzed with respect to the hip platform. Two rigid links form a P-joint (P₁) which connect the hip to the knee by two U-joints (U_H and U_K) at each end. Likewise, P₂ connects the knee to the foot by two U-joints (U_K and U_F). Two P-joints (P₁ and P₂) have reciprocating motion which means they can be actuated by a single actuator. P-joints (P₁ and P₂) provide translation along z direction. Rotation axes of U-joints (U_H, U_K, and U_F) are perpendicular and coincident. They provide rotation about x and y directions. U_H is actuated by two rotary actuators. Parallelograms (Pa) are composed of passive cables and twelve pulleys (P_H1–4, P_K1–4, and P_F1–4). Each pulley is connected to the related platforms by a passive revolute joint (R-joint). Two gimbals are designed for U_H and U_K. Also, two ball screws are used to provide reciprocating motion where the first ball screw has a hallow shaft. U_HP₁U_K leg constrains rotation of the knee platform with respect to the hip platform about the z-axis. Similarly, U_KP₂U_F constrains rotation of the foot platform with respect to the knee platform. Two parallelograms with the blue R-joints constrain rotation of the knee and foot platforms about the x-axis. Correspondingly, the two parallelograms with the green R-joints constrain rotation of the platforms about the y-axis. Since three orientation DOFs are constrained, the knee and foot of the mechanism have pure translational motion.

Fig. 1. The first conceptual design: (a) the kinematic structure and (b) the CAD model.

Details of the U-joints are shown in Fig. 2. The actuated gimbal design consists of two rotary parts (Gimbal_H1 and Gimbal_H2) which are connected to the rest of the mechanism with a rigid link (Fig. 2a). Gimbal_H1 rotates the link about the x direction where Gimbal_H2 rotates it about the y direction. Rigid link-Gimbal_H2 contact has considerable amount of friction. Although friction is a drawback of this design and several types of actuated gimbal mechanisms exist, we propose a specific gimbal design with two rotary actuators having independent motion so that each motor does not carry the weight of the other one. A closer view of Gimbal_K is shown in Fig. 2b. Gimbal_K connects the second nut to the knee and allows rotation about x and y directions. The pulleys (Fig. 2c) have same amount of rotation about the same directions as Gimbal_H and Gimbal_K: P_H1–2, P_K1–2, P_F1–2 rotate with the same amount of rotation of Gimbal_H1 whereas P_H3–4, P_K3–4, and P_F3–4 rotate with the same amount of rotation of Gimbal_H2.

Fig. 2. Details of the conceptual design: (a) Gimbal_H, (b) Gimbal_K, (c) pulley.

2.2. Conceptual design II

The kinematic structure of the second conceptual design is (UU-2Pa)-P as shown in Fig. 3a. This mechanism consists of same three platforms as a hip platform, a middle platform (knee), and a moving platform (foot). A rigid link (thigh) is connected to the hip and the knee by two U-joints (U_H and U_K). U_H is actuated by two rotary actuators. Two rigid links (shank) form an actuated P-joint (P₁) and connect the knee to the foot. U_H and U_K rotate about axes along x and y directions. P₁ allows to translate the foot with respect to the knee along z direction. Four equal long passive cables connect the hip to the knee and are anchored at points (A_H1–4 and A_K1–4). Two parallelogram loops are composed of four passive cables, the hip, and the knee. Actually, three cables are enough to form two parallelograms (one mutual cable in parallelograms). However, Fig. 3 proposes a solution with four cables. This solution includes one redundant cable. This allows to obtain a symmetrical design with two independent parallelogram loops. As long as the cables are in tension, they can be considered to be equivalent to a rigid link with a S-joint at each end. Rotation of the knee with respect to the hip about z direction is constrained by the U_HU_K leg. The parallelogram loops constrain the rotation of the knee with respect to hip about x and y directions. The hip and knee platforms that are connected with cables and the rod constitute a Wren platform [Reference Wren30] with SS, and/or US and/or UU legs [Reference Kiper and Söylemez31]. The motion of a Wren platform comprises a 2-DOFs pure translational mode thanks to the n − 1 independent parallelogram loops that are constructed by the n legs (cables and the rod in our case).

Fig. 3. The second conceptual design: (a) the kinematic structure and (b) the CAD model.

Therefore, proposed mechanism has knee and foot platforms with pure translational motion. The same Gimbal_H design is used in both designs.

A 3D CAD design of the proposed mechanism is developed in Solidworks® environment (Fig. 3b). Cables are modeled as rigid links with S-joints at each end. Four sets of cable pairs are used to form parallelogram loops, and they keep the platform parallel to each other. Since the mass of the cables and cable connection parts are very low when compared with the other components, using redundant cables results in negligible amount extra mass. Actuated U-joint (U_H) is constructed with a gimbal design (Gimbal_H) that is indicated with the orange links in Fig. 3b. Two rotary actuators indicated with black boxes actuate the Gimbal_H. Gimbal_H rotates about y direction with considerable amount of friction. A commercial U-joint product is used as a passive universal joint (U_K) and the shank is modeled as a linear actuator.

Independent motions of the leg mechanism in four different positions are illustrated as snapshots in Fig. 4 with θ _H1 (about x direction), θ _H2 (about y direction), and s (along z direction) that are actuated joint variables. Home position of the leg mechanism is shown as solid model where all joint variables are equal to zero. Red, blue, and black models show motions for θ _H1 = 30°, θ _H2 = –30°, and s = 50 mm, respectively.

Fig. 4. Snapshots of independent motions of the leg mechanism with corresponding actuation inputs: (a) θ _H1 = 30°, (b) θ _H2 = –30°, (c) s = 50 mm.

Comparing the two conceptual designs, the second design is less complicated. Since the thigh and the shank are collinear for the first design, the second design is a more human-like design. Two ball screws solution for reciprocating motion, an extra gimbal design for the U_K, and more complicated passive cable connection details make the first design less favorable. The second solution is selected to proceed with a detailed kinematic analysis as follows.

The proposed leg mechanism can be simplified as a UP kinematic chain, that is, a 2-DOFs inverted pendulum, which can rotate about x and y directions, connected to an extendable link which moves in z direction. This simplification can be made since the leg mechanism possesses pure translational motion. Accordingly, U_K joints and cable connection angles have no effect on computing the position of the foot with respect to the hip. Fig. 5 depicts a simplified model with link and joint parameters. Length of the thigh link is defined as l _T . Hip joints U_H1 (orange) rotates about x direction, whereas U_H2 (blue) rotates about y direction. Shank joint P₁ translates along z direction. The thigh link rotates about hip point (O_H) by an amount of θ _H1 about x direction and θ _H2 about y direction. The knee point O_K is at P_K(0, 0, l _T ) with respect to fixed reference frame O_H(x, y, z) when θ _H1 = θ _H2 = 0. Foot point O_F is translated from O_K by an amount of s along z direction. Shank length |O_KO_F| = s can be defined as s = l _s + Δs by considering the minimum length of the shank (l _s ). Direct kinematics formulation can be derived to compute position of O_F(O _Fx , O _Fy , O _Fz ) in O_H(x, y, z) frame depending on actuated joint angles θ _H1, θ _H2 and displacement s in the form

(1) \begin{equation}\left[\begin{array}{c} O_{Fx}\\ \\[-9pt] O_{Fy}\\ \\[-9pt] O_{Fz} \end{array}\right]=\left[\begin{array}{c@{\quad}c@{\quad}c} 1 & 0 & 0\\ \\[-9pt] 0 & \cos \theta _{H1} & \sin \theta _{H1}\\ \\[-9pt] 0 & -\sin \theta _{H1} & \cos \theta _{H1} \end{array}\right]\left[\begin{array}{c@{\quad}c@{\quad}c} \cos \theta _{H2} & 0 & -\sin \theta _{H2}\\ \\[-9pt] 0 & 1 & 0\\ \\[-9pt] \sin \theta _{H2} & 0 & \cos \theta _{H2} \end{array}\right]\left[\begin{array}{c} 0\\ \\[-9pt] 0\\ \\[-9pt] l_{T} \end{array}\right]+\left[\begin{array}{c} 0\\ \\[-9pt] 0\\ \\[-9pt] s \end{array}\right]\end{equation}

Fig. 5. Simplified kinematic model with link and joint parameters.

Velocity and acceleration level kinematics can be computed by taking derivatives

(2) \begin{equation}O_{Fx}=-l_{T}\sin \theta _{H2}, O_{Fy}=l_{T}\sin \theta _{H1}\cos \theta _{H2},O_{Fz}=l_{T} \cos \theta _{H1}\cos \theta _{H2}+s\end{equation}

θ _H1, θ _H2, and s values can be found using inverse kinematics for given position of the foot point O_F as

(3) \begin{equation}\theta _{H2}=\sin ^{-1} \frac{-O_{Fx}}{l_{T}}, \theta _{H1}=\sin ^{-1} \frac{O_{Fy}}{l_{T}\cos \theta _{H2}}, s=O_{Fz}-l_{T} \cos \theta _{H1}\cos \theta _{H2}\end{equation}

A preliminary dynamic simulation is performed to check actuation torques, actuation force, and reaction forces by using SolidWorks® Motion Analysis toolbox and presented detailly in ref. [Reference Demirel, Carbone, Ceccarelli and Kiper29]. A prototype of the proposed leg mechanism is designed.

3.1. Prototype design

The CAD model of the leg mechanism is shown in Fig. 6a. Platforms are composed of aluminum profiles. The components of the gimbal design presented in Section 2 are produced with polyoxymethylene material. Two sets of cables on two opposite sides of the platforms are used to construct the four parallelogram loops. Cable connectors are designed to guide cables with holes and pulleys (V-groove bearings). Laser cut acrylic plates are used to guide the cables and precise connection of the components attached to the hip and the knee. Since linear actuators generally cannot operate in relatively high speeds, a slider-crank linkage is used to achieve translational motion of the knee joint. In ref. [Reference Demirel, Carbone, Ceccarelli and Kiper29], there are two parallel cables as line segments on each side (totally eight cables), whereas in the new prototype there are four cables in total. The use of four cables ensures that at least two cables are always in tension. In fact, cables can only pull and not push. Accordingly, at least one redundant cable is required to ensure that if a cable at one corner is slack, the cable at the opposite corner keeps pulling. The cable redundancy guarantees the presence of at least one parallelogram loop with tensioned cables in any operation condition. This parallelogram loop – together with the rigid rod at the middle, hip, and knee – guarantees that the platforms remain parallel to each other at all times. It is worth noting that a proper operation of this redundant set-up requires that all four cables have exactly the same length of the thigh link. Therefore, the cable lengths should be carefully calibrated and tightened.

Fig. 6. Design and dimension parameters of the leg mechanism: (a) the CAD model and (b) design parameters.

Fishing lines that can carry up to 45 N are used in the prototype. A parallelogram loop composed of two cables can carry up a load up to 90 N. The translational motion between the knee and the foot was originally planned via a linear actuator, whereas this is here done via a slider-crank mechanism.

Main specifications of the prototype are listed in Table II. The 3-DOFs leg mechanism is designed for 200 mm × 50 mm step length (S_L × S_H). Stepper motors are used for the actuation. Total mass of the prototype (m_L) is 10.84 kg (6.7 kg due to actuators). Dimension parameters of the leg mechanism are listed in Table III by referring to the design parameters in Fig. 6b. Dimensions are determined according to desired step size. The thigh length (l _T ) is determined for ± 30° rotation of θ _H1. Δs _max is found as 80 mm for this step size. The crank (l _C ) and coupler (l _CO ) lengths are designed according to Δs _max, and a proper l _C /l _CO ratio is selected. The crank length is selected to be longer than half of Δs _max, because the crank cannot have 180° rotation range due to collusion with the knee platform. Passive cable system mass is about 121 g including masses of cables, guiding components, and turnbuckles. If two rigid links with U-joints at end were used instead of the four cables, there would be considerable extra weight. If the smallest available commercial U-joint in the market (DIN 808 U-joint, 50 g mass) and 6 mm steel rods (4000 series, 44 g mass) are used, the total mass would be 288 g, which is more than twice the mass of the proposed cable system. Of course, a higher number of rods with larger rod diameters could be required to carry a heavy upper structure while guaranteeing the required stiffness. Further details on the modeling and analysis of stiffness for biped walking can be found, for example, in refs. [Reference Ceccarelli and Carbone32, Reference Carbone, Lim, Takanishi and Ceccarelli33].

Table II Main design specifications of the leg mechanism in Figs. 7 and 8

Table III Design parameters (in mm) of the leg mechanism in Figs. 7 and 8

The built prototype is presented in Fig. 7. 3D-printed components are used as connection parts. Other components are selected from available products in the market. Two stepper motors (Oriental Motor Pk E 4.5A) actuate the gimbal, whereas another stepper motor (Shinano SST59D-5150) actuates the slider-crank linkage. Two magnetic encoders (Ams AS5048B) are attached to the output shaft of the actuated hip joint (θ _H1) and the crank of the slider-crank linkage. The slider-crank mechanism is used to obtain linear motion between the knee and the foot platforms. Actuator inputs are designed by using a degree 5 polynomial to get a smooth motion of the foot. A real-time PCI controller is used to apply velocity profile control and to collect data from magnetic encoders. The prototype costed about 1000 €, namely 585 € are for the stepper motors and drivers, 50 € are for the encoders, 15 € are for the 3D-printed components, 350 € are for the off-the-shelf components such as the sigma profiles, steel rods, cars for the linear guides, bearings, cardan joints, fasteners, shaft couplings, turnbuckles, and other elements for rigid connections.

Fig. 7. The built prototype of the leg mechanism for validation tests.

Turnbuckles are used to calibrate the tension of the cables. A calibration process is performed with the help of a water gauge to make all cables in same length and tension. The calibration process is repeated to ensure that the cables do not get loose during the motion and also verify that the knee platform remains parallel to the hip platform at several configurations of the mechanism.

3.2. Experimental validation

The proposed experimental set-up of the leg mechanism is shown in Fig. 8. It consists of a single leg that is tested for off-ground walking tests on the sagittal plane. Side motion tests cannot be conducted with the proposed experimental set-up, since side motion planning requires the coordination of both legs. Accordingly, the proposed side by side tests are not feasible with the proposed experimental set-up, and they might only be considered for future studies after developing a new experimental set-up with two legs. A real-time PCI card (Humosoft MF 624) is used as a controller. Stepper motors are actuated by two types of stepper drivers (DQ258M and DM278M). The black fixed frame is used for fixing the hip platform and the stepper motors of the gimbal. The target trajectory is fixed on the frame, and a laser pointer attached to the foot is used to visualize the foot trajectory. Open loop velocity profile controller is modeled in Matlab/Simulink® environment.

Fig. 8. Experimental set-up of the leg mechanism prototype in Fig. 7.

Trajectory of the foot point is adjoining of half of an ellipse, and a straight line (Fig. 9a). 200 mm × 50 mm (S_L × S_H) walking gait is planned for 8 s step time. The half ellipse and straight-line segments correspond to swinging and stance phases, respectively. The foot point tracks A-B-C-A sequence with 2 s time in the AB and CA segments and 4 s in the BC segment. 8 s step time is slow for a leg mechanism, but the prototype focuses on capability of the mechanism to accomplish desired walking gait. Low project budget and applied open loop control system are also reasons for the slow motion. The actuator inputs for the walking gait in sagittal plane are planned by using a degree 5 spline. In addition to the inverse kinematics equations found in Section 2, the crank angle θ _S should be computed for the desired s as

Fig. 9. The planned walking gait for the validation tests: (a) trajectory of the foot point in sagittal plane (YZ-plane) and (b) velocity profiles of the motors.

(4) \begin{equation}\theta _{S}=\cos ^{-1} \frac{{l_{C}}^{2}+s^{2}-{l_{CO}}^{2}}{2l_{C}s}\end{equation}

The initial position of the leg mechanism is point A. The motion of the actuated joints (θ _H1 and θ _S ) is computed in three segments as AB (0–2 s), BC (2–6 s), and CA (6–8 s). The mechanism stops at points A, B, and C; therefore, velocity and acceleration values are zero at these points. Velocity profiles of the motors for a single step are shown in Fig. 9b. The other actuated hip joint remains stationary (θ _H2 = 0) during the motion. Motion of this joint is required for side walking and diagonal motions. Although presented mechanism is designed to achieve biped walking, this is kept out of the scope of this paper.

The leg mechanism prototype is operated for the desired walking gait to test the trajectory tracing performance. Trajectory of the experiment is traced three times (three steps). PCI controller is run at 10 kHz, and the input data are generated accordingly. Angular displacement data of the actuated shafts are collected using rotary magnet encoders. Proportional gains are added to the input velocity profiles with respect first measurement, and positioning errors are decreased. Experiment is repeated three times. Snapshots of the first step are shown in Fig. 10. A prototype video taken during the tests is provided as a supplementary material.

Fig. 10. Snapshots of a validation test using the test set-up in Fig. 9.

The desired and measured positions of the actuated joints angles (θ _H1, θ _S , θ _H1mes, θ _Smes) for three steps of walking are shown in Fig. 11. The desired joint angles are calculated by using the inverse kinematics equations (3) and (4). Maximum errors between the desired and measured angles for the hip joint are Δθ _H1M1 = 0.93°, Δθ _H1M2 = 1.28°, and Δθ _H1M3 = 1.29° for measurements 1–3, respectively. Maximum errors of the slider-crank linkage are Δθ _SM1 = 1.4°, Δθ _SM2 = 0.86°, and Δθ _SM3 = 0.82°. Repeatability of the motion is good.

Fig. 11. Desired and measured angular displacements of the actuated, (a) hip joint (θ _H1) and (b) slider-crank linkage (θ_S).

Desired and computed positions are shown in Fig. 12. Sagittal plane motions of the foot are shown in Fig. 12(a). Fig. 12(b) shows y- and z-axis errors (mean of all experiments) based on encoder data. The maximum positioning error is 9.65 mm in y direction and 3.81 mm in z direction. The mechanism can accomplish BC and CA segments successfully but there are relatively large errors during the AB segment. Vibrations have considerable effect on positioning errors. Since positioning errors are computed by using kinematic equations and collected sensor data are taken from magnetic encoders, the computed errors seem to be higher than errors seen in Fig. 10 and the video (supplementary material). The computed errors including manufacturing and assembling errors can produce significant difference between designed and real kinematic parameters.

Fig. 12. Desired and computed positions of the foot point according to encoder data, (a) sagittal plane, (b) y-axis (horizontal), and z-axis (vertical) errors.

The obtained results demonstrate the engineering feasibility of the proposed design as a suitable solution for achieving static walking with only three motors per leg instead of the six motors commonly required for a legged locomotion. This allows a reduction of the robot’s own weight as well as a strong reduction of the complexity for the control of the locomotion.

It is to note that the low available budget has limited the manufacturing quality resulting in limited accuracy in path following tasks. Improving the manufacturing and using a close loop control in combination with proper sensors and servomotors will be planned as future work once a full biped prototype can be made available. Similarly, biped walking tests for straight walking, side walking, and crossing over obstacles are planned as future studies.