## 1 Introduction  
High aspect ratio aspheric workpieces are widely used in precision systems in aerospace, nuclear physics and other fields, such as rocket engines and spaceships [1; 2]. The dimensional accuracy of the high aspect ratio aspheric workpiece is important since the geometric errors would greatly influence the performance reliability of the systems [3; 4]. For example, in the rocket engine nozzles, the surface error and geometric error above the micrometer level will affect the stable flow state of the tail flame, resulting in the degradation of the engine performance [5]. A high aspect ratio aspheric workpiece is mainly produced by grinding or lapping processes [6; 7]. Due to its high aspect ratio profile, it is normally a challenge to obtain high accuracy only with the manufacturing process without measurement since the control of multi-motion axes of the machine tool to track a high aspect ratio and large amplitude profile is always suffering from low accuracy compared with that of a smooth profile [8]. Therefore, to ensure the geometric accuracy of the workpiece, it is necessary to accurately evaluate the geometric shape of the high aspect ratio aspheric workpiece [9], especially its inner profile, which is normally difficult to measure, while the outer surface profile can be relatively easily probed by many existing methods. Many efforts have been carried out to evaluate the geometric profile of aspheric workpieces. Some studies based on the coordinate measuring machine (CMM) have been utilized for dimensional measurement [10; 11; 12; 13]. Although CMM with a long tactile probe or state-of-art technique can perform a profile measurement of the high aspect ratio aspheric surfaces [14; 15; 16; 17], it is a cost of capital to prepare such an instrument for the routine inspection of the high aspect ratio aspheric part. In addition, it is burdensome to reform a CMM for on-machine measurement for high aspect ratio aspheric surfaces with different geometry characteristics. The ultrasonic method is commonly used for thickness measurement [18; 19]. Nevertheless, the ultrasonic method cannot measure the inner profile directly and accurately since its measurement resolution is generally on the order of just a few tens of micrometers, which is not suitable for precision measurement, which requires micrometer uncertainty. Laser triangulation sensors and the fringe projection profilometry method are also applied to evaluate the geometric form of the aspheric workpieces [20; 21]. It is also noted that the V-block method can be applied to evaluate the roundness and waviness profiles of the cylindrical parts in industrial conditions [22]. These methods are effective for the outer profile measurement but cannot measure the entire inner profile of the high aspect ratio aspheric workpiece because the detectable angle of the optical measuring methods is limited [23]. Motivated by the limitation of capability and efficiency to measure the high aspect ratio aspheric workpiece, a measurement method of utilizing two displacement gauges on a four-axis motion platform to evaluate the entire inner profile of a high aspect ratio aspheric workpiece with a steep sidewall is proposed. Differing from traditional methods that can only measure the upper or the bottom part of the inner profile of the high aspect ratio workpiece, this paper proposed a novel method for measuring the whole inner profile and evaluating the geometric parameters, such as the radius of the bottom spherical shape and the steepness and straightness of the sidewall, with a two-probe measurement setup. The characteristic of the proposed measurement method also enables the ability to evaluate the waviness profile and roundness of the inner and outer surfaces. The key advantage of this method is that the measurement can be performed without significant investment in time and capital to assess the profile of the high aspect ratio workpiece accurately with high compatibility for on-machine measurement.