Robotic Knee Prosthesis with Cycloidal Gear and Four-Bar Mechanism Optimized Using Particle Swarm Algorithm

Abstract

A powered transfemoral prosthesis is needed as people with transfemoral amputation show 60 percent extra metabolic cost when compared to people with no amputation. Recently, as illustrated in the literature, the most high-torque robotic knee prosthesis utilize harmonic reducers. Despite the advantage of high reduction ratio and efficiency, the harmonic drive cannot be back-driven. Therefore, the harmonic drive is not an optimal solution for prosthetic systems with direct and indirect contact with the environment. In this paper, we outline an initial design of robotic knee prosthesis. The proposed robotic knee prosthesis consists of BLDC motor, cycloidal gear with reduction ratio 13:1, four-bar mechanism, and timing belt transmission with 4:1 reduction ratio. To optimize the torque transmission and range of motion (RoM), a multiobjective optimization problem must be undertaken. The end-effector motion depends on each bar length in the four-bar mechanism. The four-bar mechanism was optimized using particle swarm optimization (PSO). To complete the optimization, a set of 50 steps was collected using wearable sensors. Then, the data of sagittal plan were processed to identify the target profile for PSO. The prototype’s computer-aided manufacturing (CAM) was completed using a MarkTwo 3D printer with carbon fiber composite. The overall design can achieve a maximum torque of 84 N.m. However, the current design lacks the elastic component (no spring is added on the actuator output), which is necessary for a functional prosthesis; this limitation will be addressed in future study.

1. Introduction

The knee complex is a limited condyloid joint. It consists of two joints with three degrees of freedom (DoF) (tibiofemoral and patellofemoral joints). Tibiofemoral joint functions mostly as a hinge joint with a slight rotation (i.e., adduction/abduction and internal/external), whereas patellofemoral joint main function is as a knee extension with the mechanical advantage of increasing the leverage of patellar tendon, which maximizes the knee torque [1]. Knee full-range of flexion is between 130° and 160°. However, in daily activities, the range drops to a value between 60° and 70° when walking, 80° while ascending stairs, and 90° from sitting to standing. Moreover, knee full-range of extension is 5° [1]. The knee joint’s peak moment occurs at early stance [2]; the knee complex generates from 0.3 to 0.7 and 1.2 to 1.7 N.m/kg when walking and running, respectively. By establishing the requirements of torque and RoM, we can discuss the weight limitation for a functional robotic knee prosthesis.

Subatmospheric suspension can provide a vacuum range between 0 and −8 inHg. During the swing phase, the fictitious force acts on the socket [3]. Therefore, the momentum should be minimized by decreasing the prosthesis mass. The knee joint prosthesis weight would be between 0.5 and 1.6 kg. However, the size of the knee prosthesis is not critical for the design configuration.

In summary, a functional design of a single DoF robotic knee prosthesis must mimic the complementary function of the knee joint. The knee joint stiffness is high during stance phase and low during swing phase. Moreover, the knee net mechanical power increases dramatically with ambulation speed and during terrain changes. A functional robotic knee prosthesis should not only be able to provide the nonlinear stiffness, but also should be able to produce the necessary amount of torque for all types of daily activities.

All robotic prostheses with an electrical actuator are adopting permanent magnet motors as they are more efficient and have high-torque density [4]. The DC motor was selected because it has a linear relationship between torque and current, which makes the control system simple and inherently stable [5]. From

Table 1. Powered ankle–foot prosthesis actuator design parameters.

Motor	Motor’s Power (Watt)	Elastic Element(s)	Stiffness
DC motor	150	Series spring	Flexion 300 kN/m Extension 600 kN/m	[9]
DC motor	150	Series and parallel springs	1200 kN/m (series) 770 kN/m (parallel)	[10]
BLDC motor	200	Series and parallel springs	600 kN/m (series) 630 Nm/rad (parallel)	[11]
BLDC motor	200	Series and parallel springs	1200 Nm/rad (series) 533 Nm/rad (parallel)	[12]
BLDC motor 1	400	N/A	-	[13]
BLDC motor 1	600	Series spring	378 kN/m	[7]
DC motor	150	Series and Nonlinear keel springs	32 kN/m (series)	[14]
DC motor	150	Series spring	50 kN/m (series)	[15]
DC motor	83	Series springs	500 kN/m (series) 200 kN/m (toe-spring)	[16]
BLDC motor	50	N/A	-	[17]
DC motor	150	N/A	-	[18,19]
BLDC motor	200	Parallel spring	43 Nm/rad	[20]
BLDC motor	200	Parallel spring	240 Nm/rad	[21]
DC motor	150	Series spring	26.6 Nm/rad	[22]
BLDC motor	200	N/A	-	[23]
DC motor	60	Series springs	120 kN/m (series) 300 kN/m (toe-spring)	[24]
DC motor	60	Series springs	60 kN/m (series) 300 kN/m (toe-spring)	[25]
BLDC motor	50	Series springs	180 kN/m(series) 300 kN/m (toe-spring)	[26]
DC motor	60	Series spring	132 kN/m	[27]
DC motor	60	Series and parallel springs	130 kN/m (series) 270 Nm/rad (parallel)	[28]
DC motor	90	Nonlinear parallel spring	No information given	[29,30]
BLDC motor	200	Series spring	445 kN/m	[31]
BLDC motor	283	Shock-absorber	No information given	[32]
DC motor	150	Series spring	208 kN/m	[33]
DC motor	150	Series springs	210 kN/m 42 Nm/rad (toe)	[34]

¹ outrunner.

, the actuators with BLDC motors can produce higher torque. However, the BLDC motor is not optimal for the application as the motor is not in a continuous high-speed operating mode [6].

As shown in Table 1 and

Table 2. Transfemoral robotic prosthesis actuators.

Motor	Motor’s Power (Watt)	Elastic Element(s)	Stiffness
BLDC motor	200	Series spring for ankle actuator	38 kN/m	[35]
DC motor	150 (extension) 60 (flexion)	Series springs	160 Nm/rad 137 Nm/rad	[36]
DC motor	150	Series spring	200 Nm/rad	[37]
BLDC motor	200	No information	-	[38]
BLDC motor	200	Series springs	385 kN/m (extension) 338 kN/m (flexion)	[39]
BLDC motor	600	Series spring	378 kN/m	[7]
BLDC motor	483	N/A	-	[40]
BLDC motor	206	Torsion series springs	1146 Nm/rad	[41]
BLDC motor	400	Torsion series springs	600 Nm/rad	[8]
BLDC motor	410	-	-	[42]
BLDC motor	40	Series springs	17–974 Nm/rad	[43]
BLDC motor	240	-	-	[44]
BLDC motor	90	Shock absorber	No information given	[45]

, most of the prototypes used a motor with 200 Watt nominal power. In recent years, more designs are using high-torque outrunner BLDC [7,8]. The new generation of powered prostheses can allow users to ambulate with a wider speed range and enhanced RoM, because the high-power motor can support the body during dorsiflexion.

Nonetheless, the utilization of BLDC motor is justifiable for the high torque-to-weight ratio. Finally, the PMSM can be potentially the best machine to develop actuators for robotic prostheses based on the motor characteristic [6].

As shown in Table 1 and Table 2, the permanent feature of all prostheses is the elastic element(s), which can help in optimizing the device power production and consumption [46]. Furthermore, the elastic element of the design reduces the shocking load in every heel-strike event. Therefore, the elastic actuators are featured consistently in the field of robotic prostheses. The actuator output is connected to the end-effector via a mechanism that amplifies the torque and generates limits on the RoM.

In Table 1 and Table 2, the consistency of the motor selection is noticeable. However, the elastic element of the actuators varies between 38 and 1200 kN/m [12,35]. The variation of the elastic elements can be attributed to the difference in working mechanisms and actuator structures. The ankle–foot stiffness is an important factor in ambulation metabolic cost [47]. In another study, it was found that the self-selected stiffness tends to increase the kinematics correlation between the biological limb and prosthetic limb with no noticeable effect on the metabolic cost [48]. Moreover, the increase in the stiffness above the self-selected stiffness led to reduction in metabolic cost and reduction in kinematic correlation [48]. This finding can answer the issue raised in [49], where the powered prosthesis’ positive mechanical work is not the only source of metabolic cost.

The vast majority of powered prostheses use linkage mechanisms, and the stability of the linkage mechanisms has been well established through analysis [50,51]. The most common linkage mechanisms are slider-crank [52] and four-bar mechanism [8]. Other designs applied multistage transmission [41,53]. The nonlinear profile and variable transmission ratio are important factors on the mechanism selection.

Table 3. Mechanisms commonly used in powered prostheses.

Common Mechanism	Advantages	Drawbacks
slider-crank	(1) simple mechanism (2) can utilize linear actuator	(1) nonlinear kinematics require more sensors for measurement	[12,16,18,31,35]
four-bar linkage	(1) more parameters can be optimized to achieve enhanced performance (2) more stable structure (3) can utilize linear actuator		[8,13,54,55,56]
direct drive	(1) can provide the most stable structure (2) can reduce the device size	(1) elastic element design is critical (2) high-torque motor is required	[21,38,41,57] [8] [42]
cable-driven	(1) design can be upgraded to two DoFs system (2) can increase maximum torque generation	(1) nonlinearity and unmodeled elasticity can affect the control behavior (2) sacrifices the design stability	[56,57,58,59,60,61]

shows the commonly used mechanisms in robotic prostheses design; some systems were developed with the direct drive method. In order to achieve the design requirements, high-torque outrunner motors were used with harmonic gear, which can achieve a high transmission ratio within the weight limit [8,41,42].

The most critical challenge for robotic prostheses’ mechanical systems is the high torque-to-weight ratio required to support the body weight. Therefore, a high efficiency speed reducer with compact design and high reduction ratio is needed. Some recent designs [8] use harmonic gear as the system reducer. However, the selection of the series elastic element is extremely critical to avoid damaging the gear box as a result of repetitive shock loads. Despite the success in utilizing cycloidal gear boxes in machining robots (high torque with direct contact with working environment) [62], a review of the literature illustrates the gap in robotic prostheses design [63]. Therefore, the main objective of this paper is to present a distinguished design of robotic knee prosthesis; in other words, the design is the first robotic prosthesis with a cycloidal gear drive and optimized four-bar mechanism. The optimized four-bar mechanism restricts the robotic knee prosthesis’ RoM, which provides structural safety in both passive and active operational modes.

The rest of this paper is arranged as follows: the design methodology is identified in Section 2. Robotic knee prosthesis mechanical design is provided in Section 3. Finally, the conclusion and future work are outlined in Section 4.

2. Method

Cycloid drives are compact high-ratio speed reducers with a reduction ratio around 10:1, which can have an efficiency up to 70 percent [64]. This is due to the friction at the contact points, which is the main source of losses in the cycloid drive [65]. The cycloidal gear efficiency can increase up to 90 percent if roller bearings are placed at the contact points [66]. Utilizing cycloidal gear in robotic prostheses is recommended because of its ability to be back-driven and due to the high torque-to-weight ratio required in the application [67]. A six-step guide line for performance testing is given in [68]; the method can be used to evaluate cycloid drive in high-performance robots and can be edited to assess the robotic knee prosthesis.

The basic method to generate the cycloidal profile is given in [69,70]. The cycloidal profiles (i.e., epitrochoidal and hypotrochoidal) were analyzed extensively in the literature [71]. In this work, the epitrochoidal profile was selected because the overall gear design can give good performance for different performance targets [72].

The PSO algorithm was introduced in [73], which was inspired by bird flocks searching for corn. PSO is widely used in unstructured continuous/discrete, multivariable, constrained and unconstrained optimization problems [74]. Figure 1 illustrates the original PSO algorithm, where i is the number of iterations.

The PSO was advanced and hybridized with different algorithms to enhance the converging speed and generalization ability [75,76,77]. In the past decades, the PSO algorithm was implemented to solve multivariable optimization problems and it exhibits good performance [78]. In [79], the authors used PSO to optimize four-bar linkage joint clearance. Several successful prototypes presented in the literature utilized linkage mechanisms to mimic the knee joint [80,81,82,83]. Linkage mechanisms were adapted for their ability to produce a nonlinear profile and amplify the input force [84]. To evaluate the mechanism in the proposed robotic knee prosthesis, the four-bar mechanism was evaluated based on the method presented in [85]. The robotic knee prosthesis scheme is outlined in the next section.

3. Mechanical Design for a Robotic Knee Prosthesis

In this section, an outline of the robotic knee prosthesis design and hardware assembly is presented. This section is divided into three subsections: design of cycloidal gear, kinematics analysis of four-bar mechanism, and system assembly. The system utilizes an outrunner BLDC motor because of the compact size and high torque-to-weight ratio; the motor maximum output torque is 0.94 N.m.

3.1. Cycloidal Gear for Robotic Knee Prosthesis

The cycloidal gear is based on an epicycloid, in which the original profile of the cycloid disk is given in [86]. The profile can be given in the Cartesian coordinate system, as shown in Equations (1) and (2), where d is the base circle diameter, ϕ is a free variable that takes value from 0 to 2π, e is the disc eccentricity, n is number of lobes (see Figure 2), ε is the radius of roller, and Γ is the contact angle between lobe and roller. To reduce cycloidal gear vibration, a cycloidal disc pair is designed with a 180 degree shift [87,88]. Figure 2 shows the profile of the cycloid disc with two reference circles.

$$x=\frac{d}{2}·\cos \left(\varphi \right)+e·\cos \left(\varphi ·\left(n+1\right)\right)-\epsilon ·\cos \left(\varphi +\Gamma \right)$$

(1)

$$y=\frac{d}{2}·\mathit{sin}\left(\varphi \right)+e·\mathit{sin}\left(\varphi ·\left(n+1\right)\right)-\epsilon ·\sin \left(\varphi +\Gamma \right)$$

(2)

$$\Gamma =arctan\left(\frac{\mathit{sin}\left(n·\varphi \right)}{\frac{d}{2ne}+\mathit{cos}\left(n·\varphi \right)}\right)$$

(3)

To assure that the system is working at its highest efficiency, the reduction ratio (i) is bound to be smaller than 20 while taking into consideration that the design’s torque requirement is around 55 N.m. The first transmission is selected to be i = 12. The maximum actuator output at this stage is approximately 10 N.m, where the cycloidal drive and motor efficiencies are 0.89 and 0.98, respectively. Figure 3 illustrates the assembled system of the BLDC motor and gear.

In

Table 4. Cycloidal gear design parameters.

	Description	Value
d	circle diameter of base circle	50.4 mm
D	reference circle diameter of the fixed ring pins	54.6 mm
N	number of pins	13
n	number of lobes	12
e	eccentricity	4 mm
i	reduction ratio	$\frac{n}{N-n}=12$
e	roller radius	3.175 mm
dh	hole diameter	$d_h=2 \left(\epsilon +e\right)$ = 14.35 mm

, a complete list of the cycloidal gear parameters and description are provided for reference.

The cycloidal disks were reinforced by continuous carbon fiber to enhance the wear and tear at the contact point by increasing the disk stiffness. The reinforcement is highlighted in blue and illustrated in Figure 4.

To link the output shaft to the four-bar linkage input point, a timing belt with a 4:1 reduction ratio is used; the maximum input torque to the four-bar mechanism should be slightly more than 39 N.m.

3.2. Four-Bar Linkage Design and Kinematic Analysis

In this subsection, the design and optimization of the four-bar mechanism is discussed, beginning with the kinematic analysis to the optimization of the linkages, and at last, the output torque calculation.

There are many methods to study the four-bar linkage. The vector representation in complex domain is one of the most effective methods for kinematic analysis. Figure 5 shows a generalized structure of the four-bar mechanism.

$$\overrightarrow{r_1}+\overrightarrow{r_2}=\overrightarrow{r_3}+\overrightarrow{r_4}.$$

(4)

$$r_1\times e^{j{\theta }_s}+r_2\times e^{j{\theta }_2}=r_3\times e^{j{\theta }_3}+r_4\times e^{j{\theta }_0}$$

(5)

$$r_1\times e^{-j{\theta }_s}+r_2\times e^{-j{\theta }_2}=r_3\times e^{-j{\theta }_3}+r_4\times e^{-j{\theta }_0}$$

(6)

By isolating the vector $\overrightarrow{r_3}$ and multiplying Equations (5) and (6), we can find the relation between θ₁ and θ₂:

$$\begin{gathered} r_1^2+r_2^2-r_3^2+r_b^2+2\times r_1\times r_2\times \left(C_{{\theta }_s}\times C_{{\theta }_2}-S_{{\theta }_s}\times S_{{\theta }_2}\right)-2\times r_2\times r_b \\ \times \left(C_{{\theta }_2}\times C_{{\theta }_0}-S_{{\theta }_2}\times S_{{\theta }_0}\right)-2\times r_1\times r_b \\ \times \left(C_{{\theta }_s}\times C_{{\theta }_0}-S_{{\theta }_s}\times S_{{\theta }_0}\right)=0 \end{gathered}$$

(7)

$$A_1^{*}=r_1^2+r_2^2-r_3^2+r_b^2$$

(8)

$$A_2^{*}=-2\times r_1\times r_b\times \left(C_{{\theta }_s}\times C_{{\theta }_0}-S_{{\theta }_s}\times S_{{\theta }_0}\right)$$

(9)

$$B_1^{*}=2\times r_1\times r_2\times C_{{\theta }_s}-2\times r_2\times r_b\times C_{{\theta }_0}$$

(10)

$$B_2^{*}=2\times r_1\times r_2\times S_{{\theta }_s}-2\times r_2\times r_b\times S_{{\theta }_0}$$

(11)

Using half-angle rules, Equation (7) is transformed into a quadratic equation. With the use of the combined variables given in Equations (8)–(11), the final equation can be written, as shown in Equation (12). The parameters of Equation (12) can be rearranged to simplify the equation by combining Equations (12)–(16).

$$\left(A_1^{*}+A_2^{*}-B_1^{*}\right)\times T_{{\theta }_2/2}^2+2\times B_2^{*}\times T_{{\theta }_2/2}+A_1^{*}+A_2^{*}+B_1^{*}=0$$

(12)

$$A=A_1^{*}+A_2^{*}-B_1^{*}$$

(13)

$$B=2\times B_2^{*}$$

(14)

$$C=A_1^{*}+A_2^{*}+B_1^{*}$$

(15)

$$x=T_{{\theta }_2/2}$$

(16)

$$A\times x^2+B\times x+C=0$$

(17)

Equation (17) can be solved to find θ₂ (output angle) in terms of θ_s.

$${\theta }_2=2\times atan2\left(-B-\sqrt{B^2-4\times A\times C},2A\right)$$

(18)

Finally, the maximum output torque is given in Equation (19). A full kinetic analysis was derived in [88]. In the early stance-phase, θ₂ = 0 and θ₃ = −π/2. Therefore, the maximum torque and the robotic knee prosthesis RoM can be determined by optimizing the bars’ lengths.

$${\tau }_{out,max}=\frac{r_3}{r_1}·{\tau }_{in,max}·cos\left({\theta }_{2,stance}-{\theta }_{3,stance}-\frac{\pi }{2}\right)$$

(19)

To find the value of the four bars, an objective function is defined by Equation (18) and the target value (Equation (20)). PSO algorithm with restriction is developed to minimize the objective function. The optimization technique was run using MATLAB (2019a). Figure 6 shows the average knee joint RoM for normal walking gait.

$$f_{obj}\left(r_1,r_2,r_3,r_b,{\theta }_s\right)=Kne{e}_{rom}-{\theta }_2$$

(20)

Using the parameters given in

Table 5. Four-bar mechanism optimized parameters.

Link	Parameters
r1	30 mm
r2	56.624 mm
r3	66 mm
rb	66.193 mm
q0	0.43706 rad

, Equation (19) yields a maximum torque of 82 N.m (considering bearing efficiency is 0.95). Overall design assembly is illustrated in Figure 7.

4. Conclusions

In this paper, an overview of a robotic knee prosthesis design utilizing cycloidal gear drive and four-bar linkage mechanism is provided. The four-bar linkage mechanism is optimized using the PSO algorithm. The prosthesis developed can provide up to 82 N.m, which can support a 120 kg person; the prototype overall weight is 0.9 Kg, including two pyramid adaptors and a battery.

The current design limitation is the lack of an additional elastic element that can mimic the knee stiffness. A combination of a series of springs is a viable option to fulfill the design requirements of high stiffness during the stance phase and minimum stiffness during the swing phase. The series spiral spring(s) will be designed as we discussed in [88], and will be attached to the actuator. Furthermore, a control system should be established to have the capability to control the robotic knee prosthesis under two modes (i.e., passive and active foot).