Kinematic Optimization of a Custom Recliner Chair
Using vector loop analysis and energy conservation, determine the kinematic optimization of a reclining chair such that the reclined position does not shift the user away from a stationary table.
B A C K G R O U N D
Reclin-INC (fictional company created for this scenario) is a furniture manufacturer and retailer that aims to provide its customers with state-of-the-art reclining chairs. Our recliners are designed to accommodate both an upright (Fig. 1, left) and supine position (Fig. 1, right) for users, using a handle to switch between the two positions. To do so, the existing designs consist of a network of linkages and kinematic pairs that extend to lift the footrest when the handle is pulled up, and a separate network that angles the support for the user’s back when the backrest is pushed back. A push down on the handle while sitting up-right, returns the chair to its original position.
Fig. 1: Change in Shoulder Location between the Upright and Supine positions
Although Reclin-INC chairs are designed with the intention of providing maximum user comfort, our client has requested for the design of a custom chair. The client is unsatisfied that in the fully reclined position, objects on nearby tables are no longer in arms length, and so is seeking a solution for this inconvenience. He has also requested that the actuation of transitioning between positions be motorized. Dissecting the problem that the client has addressed, the cause has been identified to be the shift in the user’s shoulder position after the chair has been reclined. It has specifically been determined that the shoulder position of the user translates both horizontally and vertically when the chair is reclined, resulting in the difficulty in reaching objects that were previously reachable
O B J E C T I V E
The redesign of this recliner should maintain user’s shoulder position in both the upright and supine positions, along with minimizing the work required by the user to transition between positions, by incorporating a motor that removes the need for physical exertion.
Engineering specifications were generated so as to fulfill the client’s request and so as not to alter the functionality of the original design. These requirements help guide the engineering design process.
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Final and initial positions of user’s shoulder should be coincident
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Time taken to recline chair should be the same as time taken to translate the chair (to ensure first objective) to allow for a smooth motion
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Any changes made should not alter existing functionality of chair, specifically:
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Ability of chair to lift user’s feet parallel to ground and recline user’s back 150 degrees.
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Ability of chair to withstand weight of user (i.e. at least 80 kg)
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Stability of linkages in new design, so they are not susceptible to deformation or fracture
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M E T H O D O L O G Y
To cater to the client’s need, it was determined that the chair should move diagonally upward and forward, while it is being reclined, to offset the loss in height and backwards movement of the shoulder during the reclining of the chair (Fig. 2).
In order to achieve this, a four-bar parallelogram mechanism (green links 13 and 14 in Figure 3) will be added to the base (link 10 in Figure 3) of the existing design. When constrained properly, the parallelogram mechanism will ensure that the base of the chair remains parallel to the ground throughout the entire stroke of the mechanism. The stroke will be constrained so that the mechanism only moves in the upper region, in order to avoid the unpredictable behaviour at the change-point. A motor will furthermore replace the current actuation handle, and will be used to power both the reclining and the elevating mechanisms simultaneously. This ensures smooth positioning, with minimal jerking in order to maintain optimal user comfort and safety.
To optimize the linkage lengths, positions, and angles, the vector loop method was employed. To complement the analysis, all vector loop relationships were input to MATLAB code to provide rapid numerical calculations, as well as graphical confirmation of the design in the form of the stick diagrams. To minimize the energy exerted on the mechanism by the motor, a linkage optimization was completed to determine the optimal length of link 13 given the trade-off between mass and height, where mass is dependent on the material property (density) and the length of the linkage.
Fig. 2: Original Shoulder Motion Path
Fig. 3: New Design with Labelled Links
R E S U L T S
Fig. 4: New Design Shoulder Motion Path
Reclin-INC’s current mechanism displaces the user’s shoulder position 541.75 mm horizontally backward and 180.89 mm vertically downward. Therefore, the motion of the proposed parallel four-bar mechanism should end when the points at the end of the linkages are 541.75 mm forward and 180.89 mm higher than their starting position as demonstrated by Fig. 4. By iterating through the smallest link length—determined by dividing the straight-line distance between the start and end points by 2—to the largest link length—constrained by the available spacing under the chair—the minimum energy was determined to be when the link length is 300 mm. Using this link length, the new design achieves minimal shoulder position change with minimal required work while ensuring that the initial and final shoulder position of the user is coincident
Due to interest in the client’s safety and comfort when using the Reclin-INC chair, a smooth reclining motion and experience is desirable. Given the various moving subunits of the chair (footrest, backrest and parallelogram base), “smooth motion” is satisfied if the time required for each motion is the same—starting and stopping simultaneously—and if jerking due to changes in acceleration is minimal.
Since the proposed design modification involves direct connection of the input link to the backrest, the footrest and backrest are necessarily already synced with respect to each other. Based on vector loop analysis (Figure 5) of this connection, not only must the angle of the backrest (𝜃13) equal that of the input (𝜃15) , the angular velocities of the backrest and input must also be the same (𝜔13 = 𝜔15). And because the input rotation rate of link 2 is constant according to its actuating motor, the angular velocity of the backrest must also be constant, thus eliminating the risk of sudden acceleration at the user’s neck. Therefore, smooth motion simply requires that the parallelogram motion is synced to footrest input.
Since the only change made to the reclining mechanism aspect of the original design was extending the backrest link to connect it to
Fig. 5: Backrest Vector Loop Analysis
the input link by means of other linkages, the functionality of the original chair to recline 150 degrees was therefore not altered. The stability of the chair was also determined to be unaltered given that the ground link is 540 mm, meaning that the total center of mass lies within the parallelogram shown in green in Fig. 4, as it is approximately 60% of the way along the link (since 338.56/540 = 0.63) and the user's center of mass, assuming approximately half of their body length from the bottom, is calculated to be 472.43 mm which is less than the 540mm ground link length, the chair can withstand the user’s weight (80 kg) in the supine position.
C O N C L U S I O N
The proposed design minimizes the displacement of the user’s shoulder position, compensating for the 180.89 mm downward and 541.75 mm backward displacement of the original design. This is achieved using a parallelogram four-bar mechanism. The optimal input link for this mechanism was determined to be 300 mm in length, since a 300 mm link length minimizes the work required by the motor, by reducing the maximum height that the seat (and user) reaches. Furthermore, the proposed design links the actuation of the footrest (done via two Universal AC motors) to the backrest, ensuring that each component of motion is synced, as well as eliminating any physical work required by the user to achieve the supine position. Additionally, it was determined that the shift in the center of mass does not impact the stability of the design, since the center of mass in both the supine and reclined positions lie within the foundational parallelogram structure. Finally, given the maximum anticipate load of 360.3 kN, an appropriate cross-sectional area for the linkages of the parallelogram mechanism was determined to be 58.05 mm^2, assuming a material selection of AISI 1020.
Fig. 6: Final Optimized Recliner Design
