J. Eur. Opt. Society-Rapid Publ. 21, 30( 2025) 313
Fig. 2. Lever arm machine as a five crank mechanism.
To calculate the rotational speed, another method is used: The polishing velocity on each point of the work piece( all points of area A 1 in Fig. 1) iscalculatedbytworotational velocities( formula( 6))( from the tool and the work piece) and one lateral velocity( formula( 7)) from the movement of the work piece. The radius r i is the distance between the center and the specific point in meter. The number of revolutions n i is in min �1.
A simple sketch of all velocities at the work piece on the lever arm machine are shown in Figure 3. The dotted line shows the previous position on time t n-1. The velocities of aspecific point on the work piece and tool depends on the specific diameter.
The angular velocities of the joints in Figure 1 can be described as follows. What is interesting for the polishing removal is the angular velocity at point C, therefore both formulas are solved according to x c. The two angular velocities that can be controlled by motors are x qA and x qE. And the two self-adjusting angular velocities are x qB and x qD:
* * * x C ¼ xqA þ xqB; ð6Þ
*
* * x C ¼ xqE þ xqD: ð7Þ
With this sketch and all the data the relative velocity can be calculated as follows:
|
v rot ¼ 2 r i p n i
60 1000; ð8Þ
|
|
|
v xy ¼ s
t ¼
|
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx n � x n�1
Þ 2 þðy n � y n�1 Þ 2
; t
|
ð9Þ |
vx ð; y; tÞ ¼ v xy þ v rot tool þ v rot work piece: ð10Þ
The formulas can be used to determine the current status of the polishing process step, which is important for 100 % control of the work pieces. However, in order to influence the shape of the work piece, one or more parameters must be adjusted in the process. The simplest parameters are the
Fig. 3. Sketch of the velocities.
lever length( Fig. 1f) or the three spindle rotation speeds( Figs. 1A, IE, and the polishing dish spindle).
1.1 Related work
To place this work in the context of current research, a comparative review of selected studies in automated polishing and control systems is presented below. These contributions highlight how different approaches handle process modelling, error mitigation, and system automation. In the context of knowledge-based full-aperture polishing, where the relationship between relative velocity and material removal is being quantified in pursuit of a more predictable, automatable process, it is valuable to situate this effort within the current scientific landscape. A review of selected publications reveals a progressive development in the field – from manual process replication to highfidelity kinematic modeling and advanced control schemes.
One of the most relevant contributions is that of Killinger and Thiess, who demonstrate the transfer of a traditional overarm polishing process onto a six-axis robotic platform [ 13 ]. This work emphasizes full-process automation, including in-situ cleaning and interferometric feedback. Although their approach is primarily empirical, relying on the direct replication of expert-defined parameters, the resulting system achieves reproducible sub-fringelevel polishing results. The initial jump in form deviation during the system transition underscores the sensitivity of the polishing process to even minor kinematic changes – an issue also addressed in the current work.
Complementary to this, Veselý et al. develop a multibody simulation and analytical framework for describing the kinematics of polishing systems with horizontal overarm [ 14 ]. Their model focuses on the interplay between workpiece rotation, tool motion, and arm oscillation – core components also present in the lever-arm mechanism investigated here. While their study is limited to planar surfaces