Design of planar slider-rocker mechanisms for imposed limit positions, with transmission angle and uniform motion controls

Mechanism and Machine Theory 97 (2016) 85–99

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Design of planar slider-rocker mechanisms for imposed limit positions, with transmission angle and uniform motion controls P.A. Simionescu College of Science and Engineering, Texas A&M University Corpus Christi, 6300 Ocean Dr., Corpus Christi, TX 78412, USA

a r t i c l e

i n f o

Article history: Received 26 July 2015 Received in revised form 30 September 2015 Accepted 3 October 2015 Available online 7 December 2015 Keywords: Slider-rocker Limit positions Mechanical advantage Hyper-function visualization Optimization

a b s t r a c t One possible way of converting the displacement of a linear motor into oscillatory motion of an output member is through the use of the planar slider-rocker mechanisms or PRRR in short. In this paper, the optimum synthesis of the PRRR mechanism for prescribed limit positions and best transmission angle characteristics (with and without ensuring uniform input–output motion) is considered. Systematic investigations of the design space in the respective optimization problems reveal some remarkable properties of this mechanism with potential new applications. To assist practicing engineers with selecting the link lengths of a PRRR mechanism, performance charts and parametric design charts are also provided. © 2015 Elsevier Ltd. All rights reserved.

1. Introduction This paper discusses the problem of kinematic synthesizing a PRRR linkage (sometimes symbolized TRRR) for imposed limit positions of the rocker, while simultaneously satisfying good motion transmission characteristics, with or without uniform correlation between the input and output link motions. The only English language work on this subject is [1], which in turn makes reference to German publications [2] and [3]. Note that throughout these publications, a simplifying case is considered where the coupler and rocker have equal lengths. A more recent take on this subject can be found in [4] and [5], with references [6–19] being also relevant to the subject of this paper. The general PRRR linkage has numerous applications and was studied by a number of researchers in the past. However, the majority of these reports are concerned with the fully rotating crank-slider mechanism of the type used in piston compressors, vacuum pumps, sewing machines, and some punch presses. By most accounts, including IFToMM recommendations [20], the term slider-crank continues to be used interchangeably, and at times ambiguously, to designate crank-sliders (like in piston compressors), slider-cranks (like in piston engines), rocker-sliders (like in hand pumps), and slider-rocker mechanisms (like in some variable pitch propellers, and also the subject of this paper)—see [11–19] for several application examples of this latter category. The slider-rocker mechanism is one of the two inversions of the PRRR kinematic chains with translating input, the other being the oscillating-slide also known as cylinder-incline, turning-block, or swinging-block mechanisms [21,22]. The output member (i.e. the rocker) of these mechanisms can be the strut of the landing gear of an aircraft, the steering-knuckle arm of a truck or tractor, a door that opens automatically, a robotic or excavator arm, the blades of a variable pitch propeller, the wicket gates of a water turbine etc. The synthesis of the PRRR mechanism for given limit positions can be performed graphically as explained in [6,7]. However, satisfying good motion transmitting characteristics is not guaranteed, and a trial-and-error search must be performed until E-mail address: [email protected].

http://dx.doi.org/10.1016/j.mechmachtheory.2015.10.008 0094-114X/© 2015 Elsevier Ltd. All rights reserved.

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acceptable mechanical-advantage or transmission angle properties are satisfied. To avoid overloading the links and joints, it is recommended that the transmission angle (noted μ throughout the paper) remains between 45°and 135°. If gravitational or elastic restoring forces are present, deviations in excess of ±60° from the ideal value of 90° are considered acceptable [23]. Bagci [8] studied the problem of synthesizing slider-rocker mechanisms for two prescribed positions of which one is a locking position, but without transmission angle concerns. Plecnik and McCarthy [9] investigated the design of slider-rocker mechanisms for function generation, again with no transmission angle control. Farzadpour [11] studied the PRRR mechanism as pitch control for ship propellers, while Koser [10], Moubarak et al. [13,14], Chang et al. [12], and Figliolini and Rea [15] reported on the application of the PRRR slider-rocker mechanism in the construction of robotic systems. The investigation of transmission angles in various linkage mechanisms attracted more attention from kinematicians, evident from [23], which refers over one hundred publications on the subject of transmission angle. However, of the 105 references therein, neither deals with the slider-rocker mechanism. The remainder of this paper investigates the capabilities of the PRRR slider-rocker to generate, for a given slider displacement, a specified maximum rocker swing while simultaneously ensuring best transmission angle characteristics (i) without and (ii) with imposing linear correlation between input and output. For case (i), the multi-dimensional space of the optimization problem is inspected using partial-global-minimum plots [24,25], and some interesting properties are revealed. Furthermore, if there are restrictions upon the possible linear motor placement, the problem can be solved effectively by overlapping such a partial-globalminimum plot with the operating space of the mechanism and by observing how these two correlate. Case (ii) is formulated as a multi-objective optimization problem that is solved using a modified Normal Boundary Intersection method [26]. To assist practicing engineers with their designs, parametric design charts accompanied by transmission angle, input–output linearity error, and torque-to-force multiplication factor diagrams are also provided in the paper. 2. Synthesis of the PRRR slider-rocker mechanism for prescribed limit positions and optimum transmission angle Given the slider-rocker mechanism in Fig. 1 of link lengths AB, OB, and slider offset yA, a maximum angular displacement Δφ = φ1-φ0 of the rocker is required to be generated for a linear motor displacement Smax corresponding to joint A moving between the points of coordinates (xAs, yA) and (xAf, yA). Without impairing the generality of the approach, the slider displacement is assumed equal to one i.e. xAs − xAs = 1. The dimensions of the real mechanism will be obtained at the end through scaling by a factor equal to the displacement Smax of the linear motor utilized. For the correlated input–output limit positions (xAs, φs) and (xAf, φf) of the mechanism in Fig. 1, the synthesis equations are [4] 2

2

2

AB ¼ ðxAs −xBs Þ þ ðyA −yBs Þ 2 2 2 AB ¼ ðxA f −xB f Þ þ ðyA −yB f Þ ;

ð1Þ

which are equivalent to 2

2

2

2

AB ¼ xAs −2OB  ðxAs cos φs þ yA sin φs Þ þ OB þ yA 2 2 2 2 AB ¼ xA f −2OB  ðxA f cos φ f þ yA sin φ f Þ þ OB þ yA :

ð2Þ

After subtracting Eq. (2) and then substituting xAf = xAs − 1 and φf = φs + Δφ, the normalized length of the rocker results as OB ¼

xAs −0:5 xAs cos φs −ðxAs −1Þ  cosðφs þ ΔφÞ þ yA ½ sin φs − sinðφs þ ΔφÞ

ð3Þ

Y

Bj

Bf

μj

μf ϕf O

Bs

μs ϕj ϕs

θj

As

Af A S max

xAf

Sj

yA

X

x As

Fig. 1. PRRR mechanism actuated by a linear motor with a maximum stroke Smax shown in its start “s” and finish “f” positions, and in an intermediate position “j” corresponding to a current slider displacement Sj.

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It can be seen that there are infinite many combinations of rocker angle φ0, initial slider position xAs, and slider offset yA that satisfy Eqs. (2) and (3). It is therefore possible to adjust these parameters until additional requirements are fullfield, such as (a) best transmission angle properties, (b) restricted slider offset and ground joint location, (c) link-length ratios, (d) mechanical advantage, or (e) uniform input–output displacement function. Imposing the transmission angle μ to have a minimum departure from 90° over the entire working range of the mechanism will be discussed first. The additional listed requirements (b), (c), and (d) will be observed through partial minima plots and performance charts. The combined requirements to satisfy (a) best transmission angle and (e) uniform input–output motion will be tackled in the latter part of the paper as a bi-criteria optimization problem. For a current displacement xA of pin joint A, the transmission angle can be calculated with
 OB2 þ AB2 −x2 −y2 Aj A : cos μ xA j ¼ 2  OB  AB

ð4Þ

The maximum and minimum values of angle μ occur when the mechanism is in its limit positions xA = xAs, and xA = xAf (Fig. 1). Apart from these, the xA = 0 configuration – if within the working range of the mechanism – may also produce a limit value of the transmission angle [4] and should also be verified during synthesis. Note that the product OB⋅AB in Eq. (4) approaches zero and could cause a floating-point overflow only for severely suboptimal mechanisms and can be easily avoided using a suitable penalty method. In order to find the link lengths of the PRRR linkage that generates a prescribed rocker travel Δφ while simultaneously ensuring minimum deviation from 90° of transmission angle μ, the following mini-max problem in the design variables φs, xAs, and yA has been defined: minimize F ðφs ; xAs ; yA Þ ¼ max fcs ; c f ; ce g with cs ¼ j cos μ ðxAs Þj cf ¼  j cos μ ðxA f Þj j cos μ ð0Þj if xAs  xA f b 0 ce ¼ 0 if xAs  xA f ≥ 0

ð5Þ

where cs, cf, and ce are calculated using Eq. (4). Note that the term ce is evaluated only if the product xAs⋅xAf is less than zero, i.e. if pin joint A crosses the OY axis, which ensures that objective function F never exceeds unit value. Using cos μ instead of the actual transmission angle μ contributes to a fortuitous reduction of the CPU time, particularly advantageous when generating parametric design charts and partial-global-minimum plots, each consisting of thousands of points obtained through optimization. The second benefit is that in the multi-criteria optimization problem discussed later, the transmission angle and the linearity error based objective functions will both be readily normalized. When minimizing the objective function F(φs, xAs, yA), it is essential to dismiss the solutions for which the triangular loop O-A-B changes orientation. This has been done by verifying that the cross products OBs × BsAs (the slider in its initial position), OBe × Be Ae (the slider intersects the OY axis), and OBf × BfAf (the slider in its final position) have the same sign. A design variable of the Boolean type [27] can be introduced in optimization problem (5) to specify the orientation of the triangular loop O-A-B. A second Boolean design variable can be also defined to specify the direction of rotation of the rocker (either CW i.e. Δφ N 0, or CCW i.e. Δφ b 0) for the slider moving right to left. However, the additional solutions that these two Boolean variables would reveal will be simply mirror configurations of the same basic solution, as explained next. 2.1. Design space studies through partial-global-minimum plots In order to better understand the motion capabilities of the PRRR mechanism in Fig. 1, the design space of F(φs, xAs, yA) has been investigated using the method of visualizing hypersurfaces described in [24] and [25]. According to this method, the following partial-global-minima function in two variables has been defined: ! GðxAs ; yA Þ ¼ arccos global min F ðφs ; xAs ; yA Þ ; ϕs

ð6Þ

where xAs and yA are the scan variables associated with the x and y axes of a 3D plot. The partial-global minima of F with respect to φs have been evaluated numerically using Brent’s interpolating-search algorithm [28], preceded by a grid-search step. The gridsearch serves to identify a feasible region along a current cross-section through the design space of F (corresponding to parameters xAs and yA being held constant) and also ensures that the global minimum of F with respect to φ0 is indeed found for each plot point. Function G(xAs, yA) has been represented graphically (Fig. 2) for several practical values of the rocker angle Δφ i.e. 60°, 90°, 120°, 135°, 150°, 180°. For a more convenient interpretation of the plots in Fig. 2, the elevation of the level curves has been modified interactively, and subsequently, they were labeled as arccos(G) (see reference [25] where the plotting program used

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P.A. Simionescu / Mechanism and Machine Theory 97 (2016) 85–99

Δϕ=60° yA

Δϕ=90° yA

xA0

xA0

Δϕ=120°

Δϕ=135° yA

yA

xA0

xA0

Δϕ=150° yA

Δϕ=180° yA

xA0

xA0

Fig. 2. Contour plots of arccos(G) for different values of the output member stroke Δφ. The global optimum points on these graphs are marked with “+.”

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to produce Fig. 2 is described). These partial-global minima graphs indicate that the design space of function F(φs, xAs, yA,) is symmetric about the vertical line xAs = 0.5, which means that for a prescribed rocker travel Δφ, two optimum mechanisms exist, with one being the mirror image of the other one about the OY axis. Allowing angle Δφ and the cross product OB × BA to be either negative or positive would result in the level curve plots in Fig. 2 being also symmetric about the OX axis. If Δφ and OB × BA are allowed only positive values, then such a double symmetry of the design space occurs only for Δφ = 180°. 2.2. Numerical examples The geometries of nine slider-rocker mechanisms obtained via direct minimization of the objective function F are summarized in Table 1. The table includes maximum linearity error ε and minimum and maximum torque-to force multiplication function (TFMF) information which are discussed later in the paper. The same mechanisms are shown for comparison overlapped in their limit positions in Fig. 3. The PRRR mechanism with the smallest rocker angle Δφ (i.e. Δφ = 76.35°) for which rocker OB is perpendicular to the slider axis in one of the limit position is shown in red-thick lines in Fig. 3. Note that for this mechanism, the crank and the coupler have the same length (OB = AB) and that the initial angle φ0 equals 90°. Hain [1] also noticed the same correlation for Δφ = 76.35°. However, to simplify derivations, he assumed that OB equals AB for all rocker travel angles Δφ, which yielded suboptimum results compared to the ones presented here, less for Δφ = 76.35°. Later, it will be also shown that the Δφ = 76.35° mechanism exhibits the largest linearity error of all PRRR mechanisms optimized for minimum transmission angle. Because of their more likely practical application, of the mechanisms in Table 1 and Fig. 3, those ensuring rocker travels Δφ = 120° and Δφ = 180° will be discussed in more detail. Figs. 4a and b show the two optimum PRRR mechanisms that cause the rocker to rotate CCW the amount Δφ = 120° for the linear motor extending to the left such that xAs − xAf = 1. One mechanism is the mirror image of the other one and has the normalized link lengths OB = 0.37413 and AB = 0.50935. The initial rocker angles of these two mechanisms are φs = 69.503° (Fig. 4b) and φs = −9.503° (Fig. 4c), respectively. The graphs of the rocker angle φ(S), transmission angle μ(S), and first kinematic coefficient dφ(S)/dS corresponding to the mechanism in Fig. 4b are shown in Fig. 4a. The same diagrams in Fig. 4a are valid for the complementary mechanism in Fig. 4c, assuming that the displacement of the slider is recorded in reverse, i.e. from xAf to xAs. The maximum deviation from 90° of transmission angle μ of these two mechanisms is only ±31.5°. By contrast, the more commonly used oscillating-slide linkage (RPRR) exhibits at best ±60° transmission angle deviation from 90° for the same rocker displacement of 120° [29]. The first kinematic coefficient dφ(S)/dS, which is equal to the torque-to-force multiplication factor (TFMF) [8], allows one to calculate the linear actuator force P required to overcome a given resisting moment M applied at the rocker using energy conservation, i.e. P ¼ M  TFMF ¼ M  dφ=dS:

ð7Þ

In case of the optimum slider-rocker mechanisms with Δφ = 180° (Fig. 5), the maximum deviation of the transmission angle μ is still within acceptable limits i.e. 90° ± 57.32°. Fig. 5a and b show the configurations associated to the upper optimum points in the level curve plot in Fig. 2, which are mirror of each other about the OY axis. The mechanisms associated to the lower optimum points in Fig. 2 (not shown) correspond to the same two mechanisms mirrored with respect to the horizontal axis. As discussed earlier and visible in both Figs. 4 and 5, in one of the end-positions, the rocker OB of the optimized PRRR mechanism becomes perpendicular to the slider axis. In such configurations, the motion cannot be transmitted in reverse, i.e. from the rocker to the linear actuator. This property, confirmed by the kinematic coefficient dφ/dS being zero for S = 1, is exhibited by all optimally designed PRRR mechanisms with Δφ N 76.35° (Fig. 3) and can be utilized in applications where a park/safe position is desired. Examples include latch mechanisms, safety gates, valves, sun trackers for PV panels, orientation mechanisms for parabolic

Table 1 PRRR mechanism dimensions with optimum transmission angle characteristics. Δφ [deg]

xAs

yA

φs [deg]

OB

AB

90 − μ [deg]

εmax %

TFMFmin

TFMFmax

30 45 60 76.35 90 120 135 150 180

0.50000 0.50000 0.50000 0.50005 0.55000 0.63100 0.66353 0.69338 0.75000

1.866026 1.207107 0.86603 0.63618 0.56250 0.44758 0.40294 0.36276 0.28868

119.08 110.38 100.91 90.01 83.66 69.50 62.36 55.21 40.89

1.330456 0.882041 0.66126 0.51465 0.45277 0.37413 0.35254 0.33889 0.33072

1.355337 0.896723 0.66161 0.51467 0.51250 0.50935 0.50815 0.50708 0.50518

±1.99 ±4.53 ±8.21 ±13.65 ±19.02 ±31.49 ±37.91 ±44.38 ±57.32

6.57 10.25 14.13 18.84 18.59 17.43 16.58 15.56 13.00

0.3633 0.3901 0.2887 5.8E-4 2.4E-7 8.6E-9 0.00 0.00 0.00

0.6367 1.0241 1.4434 1.9430 2.2792 3.0770 3.5376 4.0711 5.5426

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P.A. Simionescu / Mechanism and Machine Theory 97 (2016) 85–99

Fig. 3. Optimum PRRR slider-rocker mechanisms in Table 1 overlapped in their limit positions. In dashed lines are shown the geometric correlations between the joint locations of these mechanisms.

antennas, yaw drives or pitch mechanisms for wind turbines, etc.—see [5] and Appendix 1. Another interesting property of the mechanisms in Figs. 4 and 5 is that they exhibit near linear input–output function φ(S) for over 75% of their working range, a desirable feature addressed separately later in this paper.

S

ϕ

dϕ dS

S

μ

μ

dϕ d S =TFMF

(b)

μ

S

S

ϕ

μ

S

(a)

(c)

Fig. 4. Kinematic diagrams (a) of the optimum PRRR mechanisms with Δφ = 120° (b) and (c).

P.A. Simionescu / Mechanism and Machine Theory 97 (2016) 85–99

ϕ

dϕ dS

dϕ =TFMF dS

91

μ S

S μ

ϕ

μ S (a)

(b)

Fig. 5. Kinematic diagrams (a) of the optimum PRRR mechanisms with Δφ = 180° (b). The complementary mirror mechanism is not shown.

2.3. Optimum transmission angle design charts In order to assist practicing engineers with selecting the link lengths of a slider-rocker mechanism that generates a prescribed rocker swing Δφ while exhibiting best transmission angle properties, the parametric design charts and performance charts in Figs. 6 and 7 have been generated. The values used to plot the curves in Fig. 6a have been obtained by repeatedly minimizing the objective function F(φ0,xAs,yA) for successive rocker angles Δφ between 30° and 210°. To ensure that the global minima of function F are indeed found, a multistart Nelder and Mead search algorithm has been employed [25,30]. Using the optimum parameters xAs, yA, and φs extracted from the design chart in Fig. 6, the normalized lengths OB of the rocker can then be calculated using Eq. (3) or extracted from the graph in Appendix A2. Then the length AB of the coupler must be calculated using either Eq. (2). Finally, the dimensions of the real mechanism are obtained by scaling it with a factor equal to Smax. The maximum deviation from 90° of the transmission angle μ of the PRRR mechanisms designed using the chart in Fig. 6 can be evaluated using the |90-μ|max line in the companion performance diagram. The same performance diagram provides the minimum and maximum values of the torque-to-force multiplication factor, and the maximum input–output motion linearity error:

ε max


   φs −φ S j  ¼ Max S j þ Δφ j¼1  n

     

ð8Þ

both evaluated for a number of equally spaced positions of the slider (0 ≤ Sj ≤ 1) equal to twice the angle Δφ in degrees. Rocker angle φ(Sj) in Eq. (8) is the solution to the trigonometric equation a cosðφÞ þ b sinðφÞ ¼ c

ð9Þ

with constants a, b, and c equal to (see Fig. 1): 2

2

2

2

a ¼ 2xA OB; b ¼ 2yA OB and c ¼ OB −AB þ xA þ yA : Eq. (9) results by first projecting over the X and Y axes the vector loop equation OB-OA = AB of the mechanism, i.e. OB cosðφÞ−xA ¼ AB cosðθÞ OB sinðφÞ−yA ¼ AB sinðθÞ

ð10Þ

ð11Þ

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P.A. Simionescu / Mechanism and Machine Theory 97 (2016) 85–99

xA0

yA

ϕ0

ϕ0 yA

xA0

ε max

ε max

TFMF

90- μ max

Δϕ

90-μ max

TFMFmax

TFMF min

Fig. 6. Chart for selecting the optimum parameters of a slider-rocker mechanism with a given rocker displacement Δφ (above), and plots of the maximum transmission angle deviation |90-μ|max, maximum linearity error εmax, and limits of the torque-to-force multiplication factor TFMFmin and TFMFmax (below). See also Appendix 2.

and then squaring and adding these two scalar equations. Specifically, the solution of Eq. (9) is:
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  φ ¼ Atan 2ðb; aÞ  Atan 2 a2 þ b2 −c2 ; c

ð12Þ

where Atan2(y, x) is the arctangent function of two arguments that uses the individual signs of x and y to determine the quadrant of arctan(y/x). 2.4. How to handle workspace limitations? In certain applications, restrictions upon ground joint location or upon the link lengths of the mechanism may be imposed, which translate in additional constraints that must be added to the objective function (5). Formulating these constraints analytically can be very tedious, however. Alternatively, designers can work out a solution to their problem interactively, by overlapping the workspace of the mechanism with the partial-global-minimum plot in Fig. 2 corresponding to the desired rocker travel Δφ. These partial-minimum plots provide an overview upon the performance of the mechanism attainable for different slider offsets yA and initial positions of the slider xs. The following steps should be followed when implementing such an approach (see also Fig. 7):

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yA

xA0

Fig. 7. Overlap of the level curve diagram of arccos(G) in Eq. (6) and of the corresponding PRRR mechanism with Δφ = 180° that has a maximum possible slider offset yA, and a transmission angle deviation from 90° of at most ±60°. The resulting parameters of the mechanism are xAs = 0.861, yA = 0.506, φs = 135.1°, OB = 0.2906, and AB = 0.7364.

max

1.) Scale the partial-minimum level curve plot selected from Fig. 2 with a factor equal to the maximum displacement Smax of the linear actuator. Conversely, the entire mechanism and any neighboring parts can be normalized such that the linear motor displacement equals one. 2.) Position the ground joint of the rocker at the origin. 3.) Represent graphically, superimposed over the partial-global minimum diagram, any surrounding object that might constrain the slider axis location, or the motion of the links of the mechanism. Note that in a practical case, in order to avoid interferences, the rocker OB or coupler AB can have shapes other than rectangular [5].

Δϕ =180°

Δϕ =150° Δϕ =135° Δϕ =120° Δϕ =90° Δϕ =60°

Fig. 8. Pareto fronts of the bi-criteria problem minimize [F(φs, xAs, yA) and εmax(φs, xAs, yA)] generated for different maximum rocker travels Δφ.

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P.A. Simionescu / Mechanism and Machine Theory 97 (2016) 85–99

4.) Draw the linear actuator axis y = yA, and position the point of coordinates (xAs, yA), i.e. the location of joint A when the slider is in its reference position. The level curve passing through this point (or an interpolation one) will indicate the maximum expected deviation from 90° of the transmission angle μ. 5.) If the transmission angle identified at step (4) is satisfactory, use the corresponding xAs and yA values to determine the initial angle φs that minimizes the objective function F with respect to φs. For example, one can tabulate F(φs) for xAs and yA held constant and extract the value of φs for which F is minimized. 6.) Determine the normalized length OB using Eq. (3) or the chart in Appendix 2. 7.) Calculate the normalized length AB using either Eq. (2). 8.) Scale the entire mechanism with a factor Smax. Fig. 7 is an example of an optimum slider rocker mechanism with Δφ = 180° that was imposed a maximum possible slider offset yA, and a deviation from 90° of the transmission angle μ of no more than ±60°.

3. Synthesis of the TRRR mechanism with best transmission angle and uniform input–output function There are a number of applications where it is desirable to ensure a near linear correlation between the input and output member displacements of a PRRR linkage mechanism. Coupled with the need for best motion transmission properties, it leads to a multi-objective optimization problem, i.e. to simultaneously minimize the transmission angle function F(φs, xAs, yA) and the linearity error function εmax(φs, xAs, yA) in Eqs. (6) and (9), respectively. Plots of the Pareto frontiers of this bi-criteria optimization problem generated for maximum rocker swing angles Δφ equal to 60°, 90°, 120°, 135°, 150°, and 180° are available in Fig. 8. These have been generated interactively using a plotting program described in [25] that implements a modified Normal Boundary Intersection method [26]. Table 2 is a summary of solutions corresponding to these Pareto fronts, generated by directly minimizing the objective function F(φs, xAs, yA) subjected to constraints εmax = 1%, εmax = 2%, εmax = 2.5%, and εmax = 5%. Of the PRRR mechanism solutions gathered in Table 2, the least practical one appears to be combination of the Δφ = 180° and εmax = 1%, which has a too large transmission angle deviation (i.e. 6.28° ≤ μ ≤128.26°). Considering their more likely practical use (see Appendix 1), the PRRR mechanisms with Δφ = 120° and Δφ = 180° rocker angle and maximum linearity errors of 1% and 5%, and 2% and 5%, respectively, will be characterized using φ(S) and μ(S) diagrams (see Figs. 9 and 10). From the study of the accumulated motion plots in Figs. 9 and 10, it can be seen that in the case of mechanisms with lower input–output linearity error ε, the rocker does not intersect the axis of the slider. Such a non-interference requirement can be explicitly imposed in applications where multiple rockers are actuated by a single sliding rod, as it is the case of arrays of sun

Table 2 Dimensions of several PRRR mechanisms with best transmission angle characteristics and maximum linearity error of 1%, 2%, 2.5%, and 5%. Δφ [deg] 60

90

120

135

150

180

εmax

xAs

yA

φs

OB

AB

90 − μ [deg]

TFMFmin

TFMFmax

0.817891 0.758263 0.733978 0.634508 0.864444 0.823779 0.805894 0.737738 0.929294 0.855939 0.842476 0.788216 1.000414 0.867619 0.854725 0.805022 1.055918 0.888535 0.865170 0.819248 1.016367 0.967423 0.923794 0.841316

0.964541 0.966749 0.965765 0.944968 0.642256 0.649143 0.650209 0.649552 0.509334 0.483519 0.485868 0.490587 0.528195 0.426250 0.428096 0.433688 0.602047 0.391077 0.379954 0.385912 0.813706 0.433356 0.368285 0.302997

76.35° 79.75° 81.19° 86.62° 65.56 68.43° 69.56° 74.15° 56.93 57.45° 58.43° 62.21° 58.24 52.05° 52.91° 56.30° 63.60 47.46° 47.42° 50.44° 80.33 45.39° 40.31° 38.51°

0.955294 0.931996 0.920410 0.864332 0.658762 0.642410 0.634956 0.599365 0.513495 0.506557 0.500511 0.473427 0.456141 0.464978 0.459509 0.434967 0.400375 0.431754 0.429479 0.406846 0.290462 0.367027 0.377432 0.374464

0.593487 0.594531 0.595742 0.589233 0.593428 0.589917 0.586802 0.578626 0.653863 0.586094 0.583472 0.572000 0.773159 0.584693 0.580857 0.568261 0.911034 0.601064 0.578076 0.564790 1.101970 0.730229 0.647979 0.552711

±17.16 ±15.06 ±14.22 ±11.40 ±28.55 ±26.59 ±25.83 ±23.10 ±40.02 ±38.09 ±37.42 ±35.00 ±48.33 ±43.81 ±43.18 ±40.96 ±58.91 ±49.51 ±48.91 ±46.91 ±83.72 ±63.60 ±60.73 ±58.76

0.9314 0.8662 0.8368 0.7010 1.3329 1.2166 1.1659 0.9479 1.7053 1.4993 1.4228 1.1074 1.8954 1.6112 1.5188 1.1485 2.0671 1.7116 1.5891 1.1601 1.8610 1.8732 1.6982 1.0681

1.0935 1.1071 1.1158 1.1687 1.7236 1.7341 1.7420 1.7994 2.5242 2.4966 2.5025 2.5582 2.9111 2.9645 2.9675 3.0198 3.2334 3.5409 3.5214 3.5681 3.8499 5.0305 5.3191 5.1074

%

1.0 2.0 2.5 5.0 1.0 2.0 2.5 5.0 1.0 2.0 2.5 5.0 1.0 2.0 2.5 5.0 1.0 2.0 2.5 5.0 1.0 2.0 2.5 5.0

P.A. Simionescu / Mechanism and Machine Theory 97 (2016) 85–99

S

ε max=1 % ε max=5 %

ϕ

95

dϕ dS

dϕ =TFMF dS

S

μ

μ

(b) S

S

ϕ

μ

μ

S

(a)

(c)

Fig. 9. Kinematic diagrams (a) of the PRRR slider-rocker mechanisms with Δφ = 120°, optimum transmission, angle, and input–output linearity error of 1% (b) and 5% (c).

tracking PV panels driven by a single sliding rod of the type disclosed in [31]. As expected, the closer to being linear the input– output motion, the worse the transmission angle it gets. 3.1. PRRR mechanism design charts for best motion transmission characteristics and linear input–output function Table 2 provides exactly calculated parameters of several optimum PRRR mechanisms exhibiting close to linear input–output transmission function. For combinations of Δφ and εmax other than those listed in Table 2, the design charts in Fig. 11 can be used.

S

ε max=2 % ε max=5 %

ϕ

dϕ dS

S

μ μ

dϕ =TFMF dS

(b) S

ϕ

S

μ μ S

(a)

(c)

Fig. 10. Kinematic diagrams (a) of PRRR slider-rocker mechanisms with Δφ = 180°, optimum transmission angle and input–output linearity error of 2% (b) and 5% (c).

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1.0% 2.0% 2.5% 5.0%

xA0

yA xA0

yA

1.0% 2.0% 2.5% 5.0%

ϕ0

|90-μ| max

Δϕ

ϕ0

|90-μ| max Fig. 11. Design charts for selecting the optimum parameters xAs, yA, and φs of a slider-rocker mechanism with rocker travel Δφ, and maximum input–output linearity error of 1%, 2%, 2.5%, and 5%. Also given are the expected maximum deviation from 90° of transmission angle μ. See also Appendix 2.

These charts are additionally useful because they provide an overview upon the effect of changing the linear motor location (i.e. of xAs and yA), or of amending some of the requirements upon the PRRR mechanism, i.e. modifying the rocker swing angle Δφ, transmission angle variation μ, or maximum linearity error εmax.

4. Conclusions The problem of converting the input motion of a linear actuator, into the rotary motion of an output member rocker using a PRRR mechanisms has been systematically investigated. This type of linkage proved capable of generating large rocker swings of over 180°, while maintaining good transmission angles characteristics and near linear input–output function. Partialglobal minimum plots of the design space, and performance charts that allow a convenient overview upon the performances of the mechanisms, together with parametric design charts and design recommendations have been provided. A number of potentially new applications of the PRRR mechanism with large rocker swing angles are suggested in Appendix 1 to the paper.

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Acknowledgements The timely reviews and suggestions for improvement of this paper by the anonymous reviewers are gratefully acknowledged.

Appendix 1 Application examples of the slider-rocker mechanism with large rocker swing angles.

Appendix 2 If the value of parameter xAs extracted from the design charts in Figs. 6 and 11 equals 0.5, Eq. (3) yields OB = 0. In actuality xAs returned by the optimization algorithm is slightly bigger than 0.5, so xAs − 0.5 in Eq. (3) is not exactly zero. For the same combination of parameters, the denominator in Eq. (3) is also very small, yielding the type of OB values plotted in Fig. A4. Once length OB is found, then the coupler length AB can be calculated using either of Eq. (2).

Fig. A1. PRRR slider-rocker mechanism with over 180° output member swing angle used in antenna or solar panel orientation mechanisms.

Fig. A2. Slider-rocker mechanism used as pitch mechanisms for aircraft or ship propellers, or for wind-turbine rotors.

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Fig. A3. Yaw drive mechanisms for applications like solar panels, antennas, surveillance cameras, spot lights, robot manipulators, excavator booms, cranes etc.

OB

1.0% 2.0% 2.5% 5.0%

Δϕ Fig. A4. Plot of rocker length OB vs. Δφ to accompany the design charts in Fig. 6 (lower curve) and the design charts in Fig. 11 (the upper group of curves).

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